Finite Geometric Structures and Their Applications (C.I.M.E. Summer Schools, 60)

A. Barlotti ( E d.)

Finite Geometric Structures and their Applications Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 18-27, 1972

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-10972-0 e-ISBN: 978-3-642-10973-7 DOI:10.1007/978-3-642-10973-7 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 ed. C.I.M.E., Ed. Cremonese, Roma, 1973 With kind permission of C.I.M.E.

Printed on acid-free paper

Springer.com

CENTRO INTERNAZIONALE MATEMATICO ESTIVO

(c.'I.M . E . ) Z°Ciclo

-

Bressanone - dal 18 a1 27 Giugno 1972

FINITE GEOMETRIC STRUCTURES AND THEIR APPLICATIONS Coordinatore: Prof. A. BARLOTTI

R. C. BOSE:

Graphs and designs.

R. H. BRUCK:

Construction problems in finite projective spaces.

Pag.

1 105

R. H. F. DENNISTON:

Packings of PG(3, q).

I!

193

J . DOYEN:

Recent r e s u l t s on Steiner triple s y stems.

!I

201

Gruppen und endliche projektive E b e nen.

It

211

J . A . THAS:

4 -gonal configurations.

I'

249

H. P. YOUNG:

Affine triple s y s t e m s .

It

265

H.

L'U'NEBURG:

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E . )

R. C.

BOSE

GRAPHS AND DESIGNS

Corso tenuto

a Bressanone

dal

18 a 1 2 7

Giugno

1972

These lectures were prepared for a C. I. M. E. advanced Summer Institute held in 1972. The undelying research was supported by the National Science Foundation Grant G P 30958-X-

R. C. BOSE

PAGE

CHAPTER I. BALANCED INCOMPLETE BLOCK DESIGNS 11. GROUP DIVISIBLE DESIGNS 111. PARTIALLY BALANCED DESIGNS AND ASSOCIATION SCHEMES IV. STRONGLY REGULAR GRAPHS AND PARTIAL GEOMETRIES V.

THE m

m

A

L CHARACmImTION THEOREM

VI. SOME SPECIAL GRAPHS VII. GRAPH IN WHICH EACH PAIR OF VERTICES IS ADJACENT TO THE SAME NUMBER d OF OTHER VERTICES 2 ~111. GEOMETRIC AND PSEUDO GEOMETRIC GRAPHS (q + 1, q + 1% 1)

BIBLIOGRAPHY

4

18 26

40 49 58

R. C. BOSE

BALANCED INCOMPLETE BLOCK DESIGNS 1. Definition of Balanced Incanplete Block (BIB) designs.

balanced incomplete block design may be defined to be an arrangement of v objects, called "treatments", into b subsets of these objects called "blocks", if the following conditions are satisfied: (i)

Each block consists of k distinct treatments.

(ii)

Each treatment occurs in r different blocks.

(iii) Each pair of distinct treatments occur together in A different blocks. The positive integers v, b, r, k, A are called the parameters of the design. These designs axe of use in carrying out statistically controlled experiments. The parameter k is called the block size; and the paramenter r, the number of replications for the design. Easy counting arguments show that the five parameters satisfy the following relations

2. Fisher's inequality for proper BIB designs. Let 0 and $ be two

distinct treatments of a BIB design. Then 9 occurs in r blocks. The A blocks in which 9 and

@

occur together must form a subset of these r

blocks. Hence (1.2.1)

Azr

When the equality holds in (1.2.1),

. it follows from (1.1.1) that

In this case, each of the v treatments occurs in each of the b blocks.

R. C. BOSE This special case i s known a s t h e "randomized block desiplll".

A BIB

design which i s not a randomized block design, i s called a proper BIB For a proper BIB design r>A.

design.

Given a BIB design, t h e incidence matrix 19 i s defined a s follows:

where n

= 1 i f t h e i-th treatment occurs i n t h e j-th block and n

ij

i f t h e i - t h treatment does not occur i n t h e j-th block. matrix.

ij

= 0

Thus, N i s v x b

Clearly

I f B' i s t h e transpose of B, then NN' i s a square matrix of order v given by (1.2.5)

NN' =

... where Iv i s t h e u n i t matrix of order v and Jv i s a square matrix of order

v each of whose elements i s unity. (1.2.6)

It i s r e a d i l y seen t h a t

det (NN') = r k ( r

-

.

It follows t h a t f o r a proper BIB design d e t (NN' ) > 0. Hence R a n k (lOIP1)

= v.

But t h e rank of 19 cannot exceed t h e number of columns i n N.

Hence b

2 Rank

B

2 Rank NN'

= v.

R. C. BOSE Hence we have Theorem (1.2.1).

or a proper BIB design, bzv .

t h e inequality

holds. This inequality i s due t o Fisher (1940). due t o Bose (1949).

The proof given here i s

For example l e t v = 16, b = 8 , k = 6 , A = 1. Then

t h e conditions (1.1.1) are s a t i s f i e d .

Nevertheless, a BIB design with

parameters 16, 8 , 3, 6, 1 cannot k i s t since Fisher's inequality i s viol a t ed

. 3.

Symmetric BIB designs.

v = b and i n consequence r = k. Theorem (1.3.1).

A BIB design i s s a i d t o be symmetric i f

We s h a l l now prove

For a symmetric BIB design with parameters v = b,

r = k, A any two blocks have exactly A treatments i n comon. I f r = A , t h e design i s a randomized block design with v = b = r = k = A and t h e theorem i s t r i v i a l l y t r u e . Now N i s a v

x

Hence we can assume r > A .

v square matrix a i d since

we have

But from (1.2.5) C1.3.1)

NNtN = { ( r

- A ) I ~+

Jvl N

= N {(r

- h)Iv +

= NNN'

.

Jvj

Again from (1.2.6) ( d e t N ) =~ det(IiIVf )

7

0

.

R . C . BOSE Hence det N # 0 which shows t h a t N i s non-singular. side of (1.3.1)

Multiplying both

from t h e l e f t by N-Iwe have

This i s equivalent t o t h e r e s u l t t o be established.

4. meters

The complementary of a BIB design.

Given a BIB design with para-

v, b, r , k, A, we can form another design, c a l l e d the, complemen-

.tary of t h e o r i g i n a l , by taking i n t h e J-th block of t h e complementary,

those treatments which do not occur i n t h e j-th block of t h e o r i g i n a l . The number of blocks and t h e number o f treatments i n t h e complementary i s t h e same a s i n t h e o r i g i n a l design. block i s v - k .

Also t h e number o f treatments i n any

Hence using t h e subscript "0" from t h e parameters of t h e

caplementary, we have

The treatment

w i l l occur i n t h e j-th block of t h e complementary, i f it

does not occur i n t h e j-th block of t h e o r i g i n a l .

Since t h e r e a r e r

blocks of t h e o r i g i n a l i n which 0 occurs, there are b - r blocks of t h e o r i g i n a l i n which 0 does not occur.

This shows t h a t

r0 = b - r

.

A p a i r of treatments 0 and @ w i l l occur together i n t h e j-th block

of t h e complementary, i f neither of them occurs i n t h e j-th block of t h e original.

But e and

a r e i n common.

@

each occur i n r block of t h e o r i g i n a l of which X

Hence t h e number of blocks of t h e o r i g i n a l i n which

e i t h e r one or both of e and $ occur i s 2 r - X .

Thus, t h e r e a r e b - 2 r + X

blocks of t h e o r i g i n a l i n which n e i t h e r e nor $ occur.

R.

C.

BOSE

We thus have the theorem' Theorem (1.4.1).

If a BIB design has the parameters v, b, r, k, A,

the complementary BIB design has the paramenters (1.4.1)

vo = v,

bo = b, ro = b-r, ko = v-k,

A0 = b-2r+h.

A given BIB design and its complementary uniquely determine each other.

For at least one of these, the block size k;v/2,

where v is the

number of treatments. Hence, when trying to construct BIB designs we need to obtain only those for which k;v/2.

5.

The residual and the derived of a BIB desi~n. Consider a sym-

metric BIB design D with parameters

where in virtue of (1.1.1), A(v-1) = r(k-1). Choose any block of D as initial block and delete frm D the initial block and the treatments contained in the initial block. We shall show that the remaining blocks form a BIB desigq, the residual design of D. Since one block and k treatments have been deleted, the number of treatments and blocks remaining is given by V*

= V-k,

b*

P

b-1.

From Theorem (1.3.1), any two blocks have exactly A treatments in common. Hence the number of deleted treatments in each of the b - 1 blocks retained is A.

Thus, the block size in the new design is k* = k-a.

Finally, we note that treatments and pairs of treatments not occurring in

R. C. BOSE t h e i n i t i a l block remain undisturbed.

Hence

The process by which t h e new design i s obtained i s called t h e process of block section, and t h e new design i s called t h e residual of t h e origin a l design.

Hence we have the theorem

Theorem (1.5.1).

Fmn a symmetric B I B design D with parameters v = b,

r = k , A we csn obtain,by the process of block section, a new BIB design D* (the residual of D), whose parameters a r e (1.5.1)

v* = v - k ,

b* = b - 1 ,

k* = k-A,

r* = r ,

X*=A.

Again s t a r t i n g from t h e synnnetric BIB design with parameters v = b, r = k,A we can delete an i n i t i a l block, and from the other blocks r e t a i n only t h e treatments contained i n t h e i n i t i a l block. retained treatments

Then t h e number of

and blocks i s given by

Also since any two blocks of t h e original design D have exactly A treatments i n common, we have i n t h e new design

Since each of t h e retained treatments i s contained once i n the i n i t i a l deleted block, and also each p a i r of retained treatments occurs once i n t h e i n i t i a l deleted block, we have i n the new design

The process by which t h e new design has been obtained i s called the process of block intersection, and the new design i s c8lled the derived

'

R. C. BOSE of the original design. Hence we have the theorem Theorem (1.5.2).

From the symmetric BIB design D with parameters

v = b, r = k,A we can obtain by the process of block intersection a new BIB design D' (the derived of D) , whose paremeters are

6. Resolvable and affine resolvable BIB designs. We shall first prove the following Lemma: Len-

(1.6.1). If a BIB design (not a randomized block design has

parameters v, b, r, k,A and the number of blocks b is divisible by r, then Fisher's inequality can be refined to

and

Let b = nr, v = nk, then

&= k

VA r--=

r-nX.

Since r > A, and r-nA is integral we have r-nA 21. Hence b-r v - l => 1

b 2- v+r-1. A BIB design with parameters v, b, r, k, X is said to be resolvable

if it is possible to separate the b blocks into r sets

R. C. BOSE

..., (Sr,l)

(so), (s~),

with each set consisting of b/r blocks, so that

every treatment occurs exactly once in the blocks of any set (Si), i = 0, 1, 2,.

.., r - 1.

The blocks of each set

(si) are

said to constitute

a complete replication. If further any two blocks of different sets or replications have the same number m of treatments in common, the design is said to be affine resolvable. '

Since b is divisible by r for a resolvable design, the inequality (1.6.1) holds.

It would be of interest to study the conditions under

which the equality holds in (1.6.1). Let the blocks belonging to the set

(si)be

where b = nr and v = nk. Let us take any particular block, say the block BO1 of the set (So). Let 11 be the number of treatments common to the ij block BO1 and the block B of the set .(si), for i = 1, 2, , r 1, ij 2 j = 1, 2, . ., n. Let m denote the mean and s the variance of n(r 1)

... -

.

quantities Lij

-

.

Each of the k treatments occurring in BO1 occurs in the design r times. If a treatment occurs in BO1, it cannot occur in the other blocks of the set (So). Hence it occurs just r - 1 times among the blocks of the sets (S1), (S2),

..., (Sr-11.

Again the k(k

- 1)/2

Hence

pairs involved in BO1 each appear X

..., (Sr-l).

the sets (s~),(s~),

Hence

- 1 times in

% E tij ( t i j - 1 ) = %(a - i ) k ( k - 1 ) .

:zf,fj =

k f(r-l)+(~-l)(k-l)1.

Now from (1.1.1)

Hence

Hence remembering t h a t b = nr, v = nk, we have

2 Since s 2 0 , we have an a l t e r n a t i v e proof of t h e inequality

If b = v + r - 1 , then s = 0. Hence

id

= m f o r a l l v a l u e s of i and j.

This shows that t h e block BO1 has exactly m treatments i n common with each block of t h e s e t s ( s ~ ) ,

..., (Sr-1 ) .

Since t h e choice of BO1 was

a r b i t r a r y , t h i s shows t h a t any two blocks of d i f f e r e n t s e t s have exactly

m treatments i n common; i . e . ,

t h e design i s a f f i n e resolvable.

Also

since m must be i n t e g r a l , k2/v i s i n t e g r a l . Conversely i f t h e o r i g i n a l BIB design i s a f f i n e resolvable then t must be constant f o r i = 1, 2, or b = v + r - 1 .

..., r -1, j = 1, 2, ..., n.

The constant value of t

our r e s u l t s i n t h e following theorem:

2

id

i s m = k /v.

ij

Hence s2 = 0

We can sum up

Theorem (1.6.1).

If a ,BIBdesign with-pargactersv, b,. r, k, A is

resolvable

If further the design is affine reeolvable, then the equality holds in (1.6.2).

Conversely if for a resolvable design, the equality holds in

(1.6.2), then the design must be m i n e resolvable. Any two blocks of an aiiine resolvable design which belong to different replications have 2

exactly k /v treatments in ccllpan.

Hence, this mnnber must be integral.

7- 'use ef idnite raowtries for %he construction of BIB designs. (a) Consider a finite projective plane of order e = pn. identify the points of the plane with treatments, plane with blocks.

We may

the lines of the

Since each line contains s + l points, through each

point pass s + 1 lines ( d = $)

, and

each pair of points is joined by

exactly one line, we get a symnetric BIB design with paramenters

f i c n ~any ~ finite d i n e plane of order s = pn, we can similarly

obtain a BIB design with parameters

by identifying the points of the plane with treatments of the design,

and the lines of the plane with the blocks of the design. (b) In the same way we can obtain BIB designs by using the Ndimensional finite projective space PG(N ,pn) and the N-dimensional n affine projective space EG(N,p ). The function

R. C. BOSE n n denotes t h e nmber of m-flats i n PG(N,p ) , where s = p

.

By taking t h e points of F ' G ( B , ~ ~a s) treatments, and t h e m-flats of

it a s blocks, we get t h e BIB design with parameters v , b, r , k, A where pn =

and

8,

The design (1.7.1) i s then t h e special case N = 2, m = 1. Again taking t h e points of E G ( N , ~a~s )treatments, and m-flats a s blocks, we g e t t h e BIB design with parameters v, b, r , k, A where s = pn and (.=a

N

,

The design (1.7.2) i s t h e special case B = 2, m = 1. (c)

BIB designs can a l s o be obtained by using non-linear curves and

R. C. BOSE surfaces. We shall illustrate this by a,single example. Let Q2 be a non-degenerate conic in ~ ~ ( 2 , 2 ~ )Then . there are s + 1

Q2

points on

where s = 2n.

that the s + 1 tangents to polarity of

Q2.

It is known [~ose(1947 a), Qvist (195211

62 all pass through a point

The remaining s lines can be divided in two sets viz.

the s(s +1)/2 intersectors each meetiw s(s

&2

in two points, and the

- 1)/2'non-intersectors, which do not meet Q2.

d

not on

&2

points.

0, the nucleus of

2

and other th&

-

Let the s2 1 points

the nucleus of polarity be called retained

Let us study the configuration of the retained points and the

non-intersectors. We shall identify our treatments with the non-intersectors and our blocks with the retained points, a treatment being contained in a block, if the corresponding non-intersector passes through the corresponding retained point.

Clearly

If a treatment is contained in r blocks, then the corresponding nonintersector must pass through r retained points.

But each non-interaec-

tor passes through s + l points, all of which are retained points.

Hence

r = s+l. Finally any two non-intersectors have exactly one point

in conunon

(which is a retained ~oint). Hence A = 1. We thus get a BIB design with parameters (1.7.6)

v = s(s -1)/2,

b = s2 -1,

r = s+l, k = 812, A = 1

where s = 2n.

8. Non-existence theorems for symmetric BIB designs. The conditions bk = vr,

A(v-1.) = r(k-1)

,

and Fisher's inequality bzv, are necessary but not sufficient conditions for the existence of a BIB design with parameters v, b, r, k, A.

We can derive further necessary

conditions in certain special cases, but no sufficient conditions are known for the general case.

We shall now prove:

For a (proper) sylmnetric BIB design in which the

Theorem (1.8.1).

number of treatments v is wen, r - A must be a perfect square. Let 19 be the incidence matrix of the design, From (1.2.6) det

(ml) = r2(r-A)V-1.

However for symmetric designs det ( m v )= (aet rj2.

-

Since v is even r2(r AIV-l can be a perfect square only if r - A is a perfect square.

This proves the theorem [Sch~tzenberger(1949)l.

For exam-

ple, the following symmetric BIB designs are impossible:

The case when v is odd requires the use of more sophisticated methods.

The first result in this direction was obtained by Bruck and Ryser

(1949), who showed Theorem (1.8.1).

If s z l(mod 4) or 2(mod 4), a necessary condition

for the existence of a projective plane of order s(or equivalently the BIB design (1.7.1) is that if p is any odd prime dividing the square free part of s, then p. f 3(mod 4 ) . Example. A projective plane of order 6 or 14 cannot exist.

Bruck and Ryser's result was generalized by Shrikhande (1950) and by Chowla and Ryser (1950). Theorem (1.8.2).

The l a t t e r proved

A necessary condition for the existence of a sym-

metric BIB design with parameter v = b, r = k, A; when v is odd i s that the diophantine equation

has a solution i n intergers, not all zero. Fram t h i s we can deduce the following result due t o ShrilEbsnde and Raghawaren (1964 )

.

Theorem (1.8.3). square free part of A .

Let u be the square free part of r - A

and t the

Then for the existence of a symetric BIB design

with parameters v = b, r = k,

it is necessary that,

( a ) for a l l o d d p r i m e s p d i v i d i n g u b u t not t ,

-

v-l ((-1) 2 tip) = 1

(b) for all odd p r b s p dividing t but not u, (ulp) = 1

(

-

( c ) for a l l odd primes p dividing both u and t , v-1 2 and t = t g ~ . - 1 uOtOlp)= 1, *ere u = u g ~

-

Here (mlp) i s the Legendre fuuction definedby (mlp) = 1, -1 or 0 according as the residue class t o which m belongs i s a quadratic residue, non-quadratic residue or the null class.

Bruck and Ryser's theorem fol-

lows as a special case from (a). For example BIB designs with the following parameters are non-existent : (i) v ~ b = 2 9 , r = k r 8 , ( i i ) v = b = 141, r = k = 21, ( i i i ) v = b = 43,

r = k = 15,

A=2, A

= 3,

A

= 5.

The non-existence of the above designs follows by using parts (a), (b), (c) respectively of 'the Theorem (1.8.3).

9. Final remarks. It is not possible to pursue in the limited space available the various methods of construction and other properties of BIB designs. The interested reader may refer to the following papers and books:

Bose [(1939),

(1942 a), (1942 b), (1947 b), (1963 a) 1, Bose and

Shrikhande (1960 a), Fisher r(1940, (1942)1, Hanani [ (1961)~ (1965)1, H a U

(1967) , Mann (1949), ~ a o[ (19461, (1961)1, Ray-Chaudhuri and Wilson (1971), Wilson ~(1972a), (1972 b) I.

CHAPTER I1 GROUP DIVISIBLE DESIGNS 1. Definition of group divisible (GD) designs. Relations between

parameters. Suppose there are v = mn objects or treatments, which are divided into m groups each with n treatments. A group divisible (GD) design is then an arrangement of the v treatments into b sets or blocks for which the following conditions are satisfied: (i)

Each block consists of k distinct treatments.

(ii) Each treatment occurs in r different blocks. (iii) Each pair of distinct treatments which belong to the same group occur together in X1 blocks, whereas a pair of distinct treatments which do not belong to the same group occur together in X2 blocks. The scheme showing the division of the treatment's into groups is called the association scheme of the design. The association scheme will usually be written as an n x m rectangular arrangement of the treatments,

R. C. BOSE i n which treatments belong t o a group appear i n t h e same column of t h e scheme. The i n t e g e r s v , b, r , k, m, n, XI,

A

2

a r e s a i d t o be t h e parameters

of t h e design. The parameters v , b , r , k, A1, t i o n s , so t h a t only f i v e are f r e e .

A2,

m, n a r e connected by t h r e e r e l a -

Clearly

Any given treatment occurs i n r blocks.

Since each of these blocks

contains k

- 1 other treatments,

ber i s 0 .

But 8 must form X1 p a i r s with each of the n - 1 treatments

there are r ( k

- 1 ) pairs

of which one mem-

belonging t o t h e same group a s 8 , and X2 p a i r s with each of t h e n(m treatments not i n t h e same group a s 8.

-1)

Hence

It i s a l s o easy t o see t h a t r:X1' 2.

':A2

'

C l a s s i f i c a t i o n of moup d i v i s i b l e design.

Let n

ij

= 1or 0

according a s t h e i - t h treatment does o r does not occur i n t h e j-th block. Then t h e matrix N=(n

i5

)

i s defined t o be t h e incidence matrix of t h e design.

From t h e conditions

s a t i s f i e d by t h e design, it i s r e a d i l y seen t h a t

according a s t h e i-th and u-th treatments do o r do not belong t o t h e same

In numbering the treatments we shall follow the convention that the th group consists of the treatments number n(t-l)+l, n(9.-1)+2,

..., n!L.

It follows irom (2.2.1) that

where I' is the transpose of N, and A and B are n x n matrices defined by

To find the characteristics roots of NN' we have to evaluate INN'

- I0 1

where I is the unit natrix of order mn.

After some reduction

we obtain

(2.2.2)

INN1-101 = (rk- 0)(rk-vA2

- O)m-l

- 1 - O)m(n-l) .

(r A

Hence we have the following theorem: Theorem (2.2.11. meters v, b, r, k, XI,

-

-

rk vA2 and r A

IfNistheincidence mtrixofaGDdesign with para-

X2, m, n, the characteristic roots of NN1 are rk,

-

with multiplicities 1, m 1 and m(n 1 Corollary. For a GD design rk-vA2 2 0.

- 1) respectively.

This follows because the characteristic roots of NN' must be nonnegative.

We can divide GD designsinto three exhaustive and mutually exclusive classes (a) Singular GD designs characterized by r = XI, (b) Semi-regular GD designs characterized by r > Al,

rk -vAp = '0,

R. C . BOSE (c) Regular OD designs kharacterized by r > A1, rk -?Ag

3.

Singular

b*, r*, k*, A*. we get v

01) designi.

> 0.

Consider a BIB design with parameters

+,

If we replace each'treatment by a group of n treatments

= d treatments divided into

+ groups, where each group---

responds to one of the original treatments. !I'm treatments belo~lgingto the same group now occur together r* times and two treatments belonging

'

.' \

to different groups occur together A* times. We thus get st QPdesign with parameters

v = nv*,. b = b*, r = r*, k = & A1 = I*,

A2 = A*,

m =

+,

n =n

which is a singular GD design since r-A1 = 0 . Conversely consider a singular GD design with parameters v, b, r, k, A1,

Let e and

A2, m, n, where r = A1.

to the same group.

@

be any two treatments belonging

Now 0 occurs in r blocks and since r = A1,

occur in each of these r blocks and nowhere else.

@

must

Hence if a treatment

occurs in a certain block, wery treatment belonging to the group occurs in that block.

Let each group of treatments be replaced by a single

treatment in the design,

then there are

+ = m treatments in the new

design, and because any two treatments belonging to different groups occur together A2 times in the original GD design, the new design is a BIB design with psrameters

We thus get Theorem (2.3.1).

If in a BIB design with parameters

e,b*,

r*, kff,

A* each treatment is replaced by a group of n treatments we get a GD design with parameters

R. C. BOSE

Conversely every singular GD design i s obtainable i n t h i s way from a corresponding BIB design. Corollary,

For a singular GD design bzm.

This follows from t h e inequality b*:v*

which holds for the BIB

design from which the GD design has been obtained.

4.

Semi-regular GD d e s i m s .

For a semi-regular GD design we have

by defini'tion (2.4.1)

r-A1>O,

rk-v

= 0.

Hence from (2.1.1) and (2.1.2) we have

We shell now prove Theorem (2.4.1).

For a semi-regular GD design k i s divisible by m.

I f k = cm, then every block must contain c treatments f o r every group. Let e treatments from t h e f i r s t group occur i n the j t h block j ( j = 1, 2,

..., b ) .

Then

since each treatment from t h e f i r s t group occurs i n r blocks, and e w r y p a i r of treatments from t h e f i r s t group occurs i n A and (2.4.3)

1

blocks.

Prom (2.4.2)

R . C. BOSE

Hence

from (2.1.1)

and (2.4.1).

Therefore,

Since e must be i n t e g r a l , k must be d i v i s i b l e by m. j

I f k = cm then

...

= c ( j = 1, 2, , b ) i The same argument applies t o treatments of eJ any other group. This proves our theorem.

It follows from Theorem (2.2.11,

t h a t f o r a semi-regular GD design

Rank NN' = v-m+l. Now b

2 Rank B = Rank NN ' = v-m+l.

If t h e design i s resolvable then t h e sum of t h e column vectors i n N , whichcorrespondto a complete r e p l i c a t i o n i s t h e v x l column vector a l l of whose elements a r e unity. b-r+l.

I n t h i s case t h e rank of N cannot exceed

So f o r a resolvable semi-regular GD design

We then have Theorem (2.4.2). v, b, r , k, X1,

For a semi-regular GD design with parameters

h2, m, n

R. C. BOSE

If the design is resolvable, the idequality can be sharpened to

5. Regular CD designs.

For a regular OD design rk-vX2 > 0.

r > A1,

Fkom Theorem (2.2.11, NN' is non-singular.

Hence

v = Rank NN' = Rank B

2

b.

If the design is resolvable Rank B

b-r+l.

Hence we have Theorem (2.5.1). V,

b, r, k,

2,A2,

For a regular GD design with parameters m, n b

2

v.

If the design is resolvable this inequality may be sharpened to

6. Necessary conditions for the existence of symmetrical regular CD designs.

A GD design is said to be symetricsl if b = v and in conse-

quence r = k. V,

Consider a symmetrical regular GD design with parameters

b, r, k, .I1, .I2, m, n, where

From (2.2.2)

lr12 = Iwl I = r2p"-1&n-1).

R. C. BOSE It follows that

$(n-l)

i s a perfect square.

Hence we have the

theorem Theorem (2.6.1).

A necessary condition f o r t h e existence of a sym-

metrical regular GD design with parameters v, b, r , k, A1, that

P-l Q

~ ( ~ i -s ~a perfect ) square, where FJ and

A2,

m, n i s

a r e given by

(2.6.3). Corollary.

I f m i s even P must be a perfect square, and i f m i s

odd and n is even then Q must be a perfect square. We give below a t a b l e of some symmetrical GD designs whose impossib i l i t y can be proved by using Theorem (2.6.1). Table (2.6.1) Some impossible symmetrical regular GD designs Ref. No.

7.

v=b

Final remarks.

r=k

m

hl

A2

a'

Further theorems on t h e impossibility of GD

designs, analogous t o t h e Chowla-Ryser (1950) and Shrikhsnde-Raghavarao (1964) theorems f o r BIB designs can be obtained by using Hilbert norm residue symbols [ ~ o s eand Connor (1952)]. For methods of constructing and additional properites of GD designs the interested reader i s referred t o Bose, Shrikhande and Bhattachsrya (1953).

R. C. BOSE CHAPTER I11 PARTIALLY BALANCED DESIGNS ANLl ASSOCIATION SCHEMES 1. Definition of p a r t i a l l y balanced association schemes and par-

t i a l l y balanced incomplete block (PBIB) designs. treatments 1, 2,

..., v a

Given v objects or

r e l a t i o n satisfying t h e following conditions is

said t o be an association scheme with m-classes: ( a ) Any two treatments a r e e i t h e r l s t , 2nd,

..., o r m-th

associates,

t h e r e l a t i o n of association being symmetrical, i . e . , i f the treatment a i s t h e i-th associate of t h e treatment

8, then 0 i s t h e i-th associate of

t h e treatment a. (b) Each treatment has ni, i-th associates, t h e number ni being independent of a. ( c ) I f any two treatments a r e i-th associates then the number of treatments which are j-th associates of a and k-th associates of 8 i s p

i

jk

and i s independent of t h e p a i r of i-th associates a and 8. The numbers (3.1.1)

v , ni,~:k

( i , j , k = 1, 2, ..,m),

a r e t h e parameters of t h e association scheme. I f we have an association scheme with m classes, then we get a PBIB design ~ 5 t hr replications and b blocks based on t h e association scheme, i f we can arrange t h e r treatments i n b blocks such t h a t (i)

Each block contains k treatments ( a l l d i f f e r e n t ) .

(ii)

Each treatment i s contained i n r blocks.

(iii)

I f two treatments a and 8 are i-th associates, then they

occur together i n Ai blocks, t h e number Ai being independent of t h e part i c u l a r p a i r af i-th associates a and 8 ( i = 1, 2,

..., m ) .

R. C. BOSE

,

For a PBIB design based on any association scheme, t h e parameters of

the scheme may be called parameters of t h e f i r s t kind, and t h e additional parameters

may be called parameters of t h e second kind. Clearly

(3.1.3)

2.

VT

= bk, n l + n 2 +

... + nm = V-1,

Relations between the p a r w e t e r a of association schemes.

definition t h e number pi

jk

c i a t e s we start with.

is independent of which pair a,

13

By

of i-th asso-

Consider t h e p a i r 6, a ; we see a t once t h a t

The following further r e l a t i o n s are easy t o prove:

Theee r e l a t i o n s were proved by Bose and Nair (19391, i n t h e i r paper introducing t h e PBIB designs.

Theee a r e all the r e a l t i o n s i n case m = 2

but f o r m+ 3 f u r t h e r r e l a t i o n s were discovered by Bose and Mesner (1959). It i s useful t o make a convention t h a t each treatment i s i t s own

zero-th associate and of no other treatments.

Then c l e a r l y we must take,

R.

C . BOSE

We can now write

for

, 5,

k

= 0 1,

... m

It should a l s o be noted t h a t (3.2.4) remains

v a l i d i f one or more of i, k, j i s zero. Also for a PBIB design based on t h e association scheme we must have

For a two class association scheme the values of t h e parameters (1, J , k = 1, 2) may conveniently be written i n t h e form of two symmetric matrices

The definition given i n paragraph 1, f o r association schemes i s not minimal, i . e . , the constancy of sane of t h e parameters can be deduced

f r o m others.

In particular f o r two class association schemes, Bose and

1 ClatKothy (1955) proved that t h e constancy of v , nl, pll,

pl:

guarantees

t h e constancy of a l l t h e other parameters of a two class association scheme..

3.

Sane examples of two c l a s s association schemes.

below some examples of two c l a s s association schemes.

We shall give

This enumeration

i s for i l l u s t r a t i v e purposes and i s not exhaustive.

designs based on them, see Bose (1963 b)

.

For examples of PBIB

( a ) The group divisible (GD) association scheme.

In t h i s case there

are mn treatments, which are divided i n t o m groups of n treatmentseqh. Two treatments belonging t o the same group are f i r s t associates, and two

treatments belonging t o different groups are second associates.

The

association scheme can be exhibited by writing down the mn tre.atJnents i n the form of a rectangular array, t h e treatments of the same group occupying the same column.

It is readily seen that the parameters of the asso-

ciation scheme so obtained are

For example, l e t m = 4, n = 3.

The corresponding GD association

scheme i s

The first associates of the treatment 1 are 5 and 9, and the second associates a r e 2, 3, 4, 6, 7, 8, 10,11, 12. GD designs have already been considered i n Chapter 11.

They are now

seen t o be a special class of PBIB designs, viz., those which are based on the GD association scheme. (b) The triangular association scheme.

We take an m x m square, and

f i l l - i n the m(m-1)/2 positions above the leading diagonal by different treatments, taken i n any order.

The positions i n the leading diagonal

are l e f t blank, while positions below t h i s diagonal are f i l l e d so that

.

R. C. BOSE

the scheme is symmetrical with respect to the diagonal.

Two treatments

in the same row (or same column) are first associates. Two treatments which do not occur in the same row or same column are second associates. It is readily verified that the parameters of the association scheme so obtained are

This scheme is called the triangular association scheme. As an illustration take m = 5.

The association scheme is then

Two treatments which are in the same row or same column are first associates.

Rro treatments which do not occur in the same row or column

are second associates. (c )

The singly linked block (SLB) association scheme.

Consider a

balanced incomplete block (BIB) design D with b treatments, v blocks, k replications, block size r andA= 1, i,e, every pair of treatments occurs in exactly one block.

Then

Consider v new treatments each corresponding to one block of D.

Two

of these new treatments will be called first associates if the corresponding blocks of D have a common treatment and second associates if the cor-

responding blocks of D have no common treatment. 'Shrikhande (1952) has shown t h a t this association r e l a t i o n s a t i s f i e s t h e conditions ( a ) , ( b ) , ( c ) of paragraph 1 with parameters,

This association scheme i s defined t o be an SLB scheme.

Every BIB

design with X = 1 gives r i s e t o such a scheme. (dl

The Latin

square

(4) association

scheme.

treatments which may be s e t f o r t h i n a k x k scheme. ,treatments a r e 1, 2,

Consider v = k

2

Thus i f k = 4 and t h e

..., 16, we have t h e scheme

For t h e case r = 2, we define two treatments a s f i r s t associates i f they occur i n t h e same row or column of t h e square scheme, and second associates otherwise. L2 association scheme.

The association scheme sodefinedmay be called t h e The parameters of the L2 scheme a r e

I n t h e general case 2 6 r < = k + l ,we take a s e t of r - 2 mutually orth-

R. C. BOSE ogonal Latin squares ( i f such a set exists).

For an Lr association

scheme we then define two treatments t o be first associates i f they occur together in the same row o r column of the square scheme, or i f they correspond t o the same symbol of one of the Latin squares. f i n e them t o be second associates.

Otherwise we de-

For example i f k = 4, r = 4 and we

take the Latin squares

C L I~

C L I~

then the first associates of the treatment 7 and 5, 6, 8, 3, 11, 15, 4, 10, 13, 1, 1 2 , 14 because the treatment 7 corresponds t o the symbol 4 in [L1]

and the symbol 1 i n [L2].

The parameters of the Lr association

scheme a r e given by

(el

The negative Latin square association scheme.

This important

c l a s s was discovered by Mesner (1967) and i s defined by the following parameters

R. C. BOSE 1 (pjk) =

(3.3.16)

( r + l ) ( r + 2 )- ( k + 2 ) (k-r - l ) ( k + l )

(k-r-l)(k+l) '(k-r -l)(k-r)

Exemples of t h i s association scheme will occur i n subsequentchapters.

4.

Association matrices.

We define

where b

= 1

i f the objects a and 8 a r e i t h associrtes

= 0, otherwise. Bi i s a symmetric matrix, i n which each row t o t a l and each column t o t a l

is n i' Among the numbers

8 only one i s unity, i.e., bai i f a and 8 a r e i-th associates.

where Jv i s the v x v matrix each of whose elements i s unity. It also follows t h a t the linear form

i s equal t o the zero matrix i f and only if

Hence

R. C. BOSE

hence the linear functions of Bo, B1, basis Bo, B1,

**

, Bm.

, B'm

*

form a vector space with

One can now p r w e [ ~ o s eand Meaner (1959)],

Lemma (3.4.1).

v

I

(3.4.4)

bayjb yk 8

jk bsa0 + * * - + i p b 6a i + * * - + p m b s jk . jk am'

sP0

Y=l

We' now note t h a t t h e left-hand side of (3.5.4) is the element i n the 8 a t h row and 8th bolumn of t h e product B 3 q , andbai i s t h e element i n the a t h row and 6th column of ~

~ = 0, (

1* * , m).

0 1 BjBk = pjkBO+pjkBl+

(3.4.5)

Thus tpm jk Bm'

The product of t m matrices of the form (3.4.31, where t h e c are scai

lars, may be expressed a s a l i n e a r combination of terms of the form B 3 9 and w i l l reduce t o the form (3.4.3). The s e t of matrices of t h i s form i s therefore closed under multiplication. ian group under addition.

It i s clear t h a t it forms an Abel-

Thus the linear functions of Bo, B1,

-..,Bm

form a ring with unit element, which w i l l be a l i n e a r associative algebra i f t h e coefficients ci range over a f i e l d . mutative

Multiplication i s also com-

.

By evaluating B.B Bk i n t m different ways using (3.4.5) it follows

3

a s a consequence of the associative low of multiplication t h a t

In these equations the summation over u runs fran O'to m and the remaining indices are arbitrary but fixed,

Rciw let us aef ine

Pk by

Now the left side of (3.4.6) is the element in the i-th row and t-th column of P P J 9' Also the element in the i-th row and t-th column of PU is t piu, so that the right side of (3.5.6) is the element in the i-th row and t-th column of

Hence we have

(1.4.7)

0 1 PJPk = pjkPo+pJkP1+ .**+pm Pm.

jk

Thus, the P's multiply in the same manner as the B's. k = i and 0 otherwise, the 0th row of

i Since pOk = 1 if

Pk contains a 1 in column k

and 0's

in other positions, which is enough to show that if C

P +c1P1+ * * * +cmPm = 0 ,

0 0

then C O P C1

i.e., Po, pl,

P

... =

m = 0 ;

C

.-.,pm are linearly independent.

They thus form the basis

for a vector space and combine in the same way as the B's under addition, as well as under multiplication. They provide a regular representation in

matrices of the algebra given by the B's, which are v x v matrices.

In

particular, Po =

Im+l.

Since the Bas are c m m t a t i v e , t h e P a s a r e canmutative. they a r e not incidence matrices and a r e not symetric. equal r o w t o t a l a , but has t h e same equal column t o t a l s

In general

Pk does not have

% 6s Bk.

lkpPlra r e equal t o n5'

ogy with (3.4.2), all elements of row j of

In analLet

B = c B + c B + * * * + cB 0 0 11 mm

9 any

element of our algebra, and l e t f (A ) be a polynomial.

Then we can

express f ( ~ =)

a 0B0 + a1B1+ * * - + LmBm'

If

P = coPo+c P

+*-•

11

+c P m m

i s the representation of B, then

f ( P ) = rOPO+alPl+'*'

+ampm.

Let f (A) be the minimum function of B and $ (A) t h e minimum function of P.

i.e.,

Then f(A) i s the monio polynomial of l e a s t degree for which

f ( ~ i)s divisible by $ ( A ) . Similarly

$(XI is

divisible by f(A).

Since both are monic polynom-

ials,

That is, B and P have t h e same d i s t i n c t characteristic roots, and every matrix B has a t most m + l d i s t i n c t characteristic roots, which are solut i o n s of the minimum equation of

P.

R. C. BOSE

5. Combinatorial applications of t h e algebra of association matrices. ( a ) Consider a PBIB design based on an m class association scheme, with the association matrices B deiinedby (3.4.1). i

be the indicence matrix of the design, i . e . n

ij

Let

= 1 or 0 according a s the

treatment i does or does not occur i n the j-th block. (3.5.2)

B = B?i' = rBO+A B +.-.+A B 11 m m'

(3.5.3)

P = rPO+A P + - * - + A P 11 m m'

Then

The elements of B?il are non-negative and for connected designs, i.e. designs i n which w e r y treatment contrast i s estimable, HH' is irreducible. Also i n v i r t u e of the identity (3.2.9) t h e sum of the elements i n w e r y row of NN' is rk.

Hence

is a stochastic matrix ( i . e . an irreducible matrix for which t h e sum of each .row i s unity).

For such a matrix [ ~ r a u e r(1952) 1unity is a simple Hence rk is a simple root

root and i s greater than all the other roots.

of B and is, therefore, also a simple root of P. One can now show

hose

(1963 b) 1t h a t the m characteristic roots of

P other than r k a r e the roots of the matrix.

where (3.5.6) 6ij

p r j = r6

ij

+ AlpilJ +

...+Ampi m - n i A i j

being t h e Kronecker delta. (b) For a balanced inccmplete block (BIB) design we have derived i n

R. C. BOSE

Chapter I, t h e inequality b 2 v due t o Fisher ( 1 9 4 0 ) ~where b i s the num-

-

ber of blocks and v i s t h e number of treatments. corresponding r e s u l t i s f o r PBIB designs.

One may ask what t h e

Now

blrank N

2 rank NN '

.

But t h e rank of 1919' i s v , unless NN' i s singular, i .e. a c t e r i s t i c r o o t , i n which case i s t i c root.

P and

, has

a zero char-

therefore P" has a zero character-

Thus:

A necessary condition f o r b z v i n a PBIB design i s

IP;~I

(3.5.7) where p*

id

= 0

is given by (3.5.6).

Thus Fisher's inequality b > v i s s a t i s f i e d i n general.

It can be

v i o l a t e d by only those designs f o r which (3.5.7) i s s a t i s f i e d .

sult i s due t o Nair (19k3).

This re-

An a U e r n a t i v e proof w i l l be found i n Bose

(1952). Bose and Mesner (1959) have given a general method of d c a a t i n g t h e m u l t i p l i c a t i e s of t h e r o o t s of B = NN1. We s h a l l i l l u s t r a t e t h e determination of t h e m u l t i p l i c a t i e s a,, = 1, al, a2 of t h e roots O0 = r k ,

el, e2,

i n t h e special case m = 2.

r o o t s of P" given by (3.5.5).

Now 0 and 0 a r e t h e c h a r a c t e r i s t i c 1 2

Setting

we f i n d a f t e r some calculation tinat

R. C . BOSE Now (3.5.9)

trI = l+al+a2 = v

(3.5.10)

trNNt = rk+a181+a2e2 = vr,

whence after some calculation [connor and Clatworthy (1954)l

It is interesting to note that the multiplicities al and a2 depend only on the parameters of the association scheme, i.e. parameters of the first kind. The corresponding general result is due to Bose and Mesner (1959). Since the multiplicities ai are expressible in terms of the parameters of the association scheme, we cannot have a set of parameters leading to nonintegral values ai., This fact can be used to prove the impossibility of certain association schemes. We are now in a position to see how Fisher's inequaulity should be modified for the case when

P and therefore P has a zero characteristic

root. Leta be the multiplicity of the root zero of B = NN'.

Then

b & rank N

& rank NIi' =v-a. Hence Fisher's inequality is replaced by [connor and Clatworthy (1954)]

6. Final remarks. For the construction and further properties of BIB designs the reader may refer to the following papers: Archbold and

R. C .

BOSE

Johnson (1956)~ Bose and Hair (1939) , Bose and Shimamoto (1952)~ Bose, Clatworthy and Shrikhande (19541, Clatworthy [ (19541, (1955), (1956)I, Maswarns C(19611, (1964 a ) , (1964 b)],

Bair C(1950, (1951 a ) , (1951 b)],

Ogawa [ (1959)~ (1960)1, Ray-Chaudhuri [(1962 b) , (1965) 1and Shrikhande (1965 1.

STRONGLY REGZTLAR GRAPHS AND PARTIAL GEOMETRIES 1. Strowly regular graphs.

A f i n i t e graph G consists of a f i n i t e

s e t of v vertices, and a r e l a t i o n adjacency such t h a t any two d i s t i n c t v e r t i c e s of G may be e i t h e r adjacent or non-adjacent.

Adjacent v e r t i c e s

may be said t o be joined and non-adjacent v e r t i c e s t o be unjoined.

W e

s h a l l be concerned with f i n i t e graphs only, and use the word graph i n t h e sense of f i n i t e graphs. m e graph G i s said t o be regular (of valence nl) i f each vertex of G i s joined t o exactly nl other vertices.

In t h i s case each vertex w i l l

be unjoined t o exactly n other vertices, where 2

A regular graph G w i l l be said t o be strongly regular i f ( i ) any two

v e r t i c e s which a r e joined i n G, are both simultaneously joined t o exactly 1 pll other vertices ( i i ) any two vertices which a r e unjoined i n G , a r e

both simultaneously joined t o exactly pl:

vertices.

A strongly regular graph G thus depends on four parameters

R. C. BOSE

(4.1.2)

V'

where n2 i s given by (8.1.1).

Dl¶

1 Pll'

2 Pll

The concept of a strongly regular graph i s

-.

isomorphic t o t h a t of a 2-class association scheme.

The v v e r t i c e s correspond t o t h e v objects o r treatments of t h e association scheme and two

d i s t i n c t v e r t i c e s of G a r e adjacent o r non-adjacent according a s t h e corresponding treatments of t h e association scheme a r e f i r s t a s s o c i r t e s or second associates.

Rote t h a t t h e constancy of

V,

1 2 nl, pll and pll guaran-

t e e s t h e constancy of all t h e other parameters of the association scheme [ ~ o s eand Clatworthy (1955)]

. '!Thus it w i l l be convenient t o c a l l two

v e r t i c e s of a strongly regular graph G f i r s t associates i f they are adjacent and second associates i f they a r e non-adjacent.

Then given any two

v e r t i c e s x and y which a r e i-th a s s o c i a t e s , t h e number of v e r t i c e s which a r e simultaneously j-th associates of x and k-th associates of y i s i pjk[i, j ,k = 1 ~ 2 1 . Also a vertex may be called i t s own O-th associate.

Hence we may allow i, j , k t o t a k e a l s o t h e value 0.

The formulae (3.2.4)

through (3.2.9) and which were proved f o r association schemes remain valid.

I n p a r t i c u l a r we have

The adjacency matrix of a graph G with v v e r t i c e s i s defined t o be t h e v x v matrix A = ( a

ij

) where a

adjacent and zero otherwise.

ij

= 1 i f t h e i-th and j-th v e r t i c e s a r e

I n p a r t i c u l a r aii = 0.

When G i s strongly

-

1 2 regular with parameters (v, nl, Plly pll)

,

and B o y BlY B2 a r e t h e associa-

t i o n matrices of t h e corresponding association scheme then B1 i s t h e ad-

R. C. BOSE jacency matrix of G.

The complementary graph

5 of

a graph G i s defined

t o be a graph with t h e same v e r t i c e s as G but with t h e r e l a t i o n of adjacency and non-adjacency reversed. adjacency matrix of

a i s Be.

When G i s strongly regular then the

W e can now define t h e matrices of Po, PI,

P2 a s i n Chatper I11 and a l l t h e formulae regarding t h e characteristic roots or eigenvalue of t h e l i n e a r functions of Po, P , P2 or Bo, B1, 1

B2

can be taken over.

2.

Seidel equivalence of strongly r e f l a r

graphs.

Let G be a

strongly regular graph with parameters

We can obtain another graph G* from it by t h e following process: Let t h e s e t of vertices V of G be divided i n t o d i s j o i n t subsets, V1 and *

V2, V = V1UV2.

G* has t h e same s e t of vertices a s G.

Two vertices of

G* both of which belong t o V1 o r t o V2 a r e adjacent o r non-adjacent

according a s they are adjacent o r non-adjacent i n G.

i n G*

Two vertices of G*

one of which belongs t o V1 and t h e other t o V2 a r e adjacent i n G* i f they are non-adjacent i n G, and non-adjacent i n G* i f they are adjacent i n G. Then G* may be said t o be derived from G by complementation with respect I f G* i s strongly regular it i s defined t o be Seidel

t o V1 and V2.

equivalent t o G, or more b r i e f l y S-equivalent t o G [ ~ e i d e l(1967)l. Let

lvll

= vl,

I V 2I

= v2, then v = v1+v2.

I n writing down the adja-

cency matrix of G, we may take t h e f i r s t vl rows (columns) t o correspond t o t h e v e r t i c e s i n V1 and t h e l a s t v2 rows (columns) t o correspond t o t h e v e r t i c e s i n V2.

Then we c a n e i t e the adjacency matrix of G as

R. C . BOSE

where A11 and AZ2 a r e square matrices of order vl and v

A12

is a v

x

1

v2 matrix, and A21 =

Ai2.

2

respectively,

Then c l e a r l y t h e adjacency matrix

of G* i s

One can now investigate t h e conditions under which G* i s strongly regular and therefore by d e f i n i t i o n i s S-equivalent t o G.

In t h i s con-

nection Bose and Shrikhande (1970) proved t h e following theorems: Theorem (4.2.1).

Let G be a strongly regular graph with parameters V,

1 nls Pll'

2 Pll'

I f t h e v e r t i c e s of G a r e divided i n t o two d i s j o i n t subsets V1 and V2, where

1 vll

= vl,

I V21

= v2, then t h e necessary and s u f f i c i e n t condi-

t i o n s f o r t h e graph G* aerived from G by complementation with respect t o V1,

V2, t o be strongly regular a r e

( a ) In G each vertex i n V i s adjacent t o wl v e r t i c e s i n V (and 1, 1 therefore nl -wl v e r t i c e s i n v2); a l s o each vertex i n V i s adjacent t o 2 w2 v e r t c i e s i n V2 (and therefore t o nl -w2 v e r t i c e s i n V ), where 1

When these conditions a r e s a t i s f i e d t h e parameters of G* a r e given by

I f t h e graph G* i s required t o have the same parameters a s G , then

R . C . BOSE I* = pll1 and pU 2* = pU. 2 This automatically ensures that pU Also v v2 1 = 2' q - w 2 = -p i.e., i n G each vertex of V1 is adjacent t o

:n = nl. nl-wl

exactly half the vertices i n V2, and each vertex i n V2 is adjacent t o exactly half the vertices i n V We therefore have 1' Let G be a strongly regular graph with parameters Theorem (4.2.2).

If t h e vertices of G are divided into two disjoint subsets V1 and V2,

,then the necessary and,mUficient conditions for the graph G* derived

frcm G by camplimentation with respect t o V1 and V2, t o be strongly regular with the same parameters a s G are ( a ) In G each v e r t u i n V1 i s auacent t o exactly half the vertices i n V2, and each v e r t u i n V2 i s sdjacent t o exactly half the vertices in

.

3.

Partial geometries and the corresmndinu PBIB d e s b s .

A par-

t a i l geanetry ( r , k, t ) i s a system of points and lines, and a relation of incidence between then.satisfying the following axioms:

Al.

Any two distinct points are 'incident with not more than one line.

A2.

Each point is incident with r lines.

A3.

Each l i n e i s incident with k points.

Ah.

I f the point P is not incident with the l i n e a , there are exact-

ly t lines (t: 1 ) which a r e incident with P, and also incident with some

point

incident with I . Clearly 1: t: k, 1; t ;r. ( a ) I f there were t w o distinct lines

and m each incident with two

R. C. d i s t i n c t points P and P then A 1 would be contradicted. 1 2,'

BOSE Hence:

Any two d i s t i n c t l i n e s a r e incident with not more than one

A'l.

point. Given a partial geometry ( r , k, t ) , there e x i s t s a rlual p a r t i a l geometry (k, r, t ) , obtained by c a l l i n g t h e points of t h e f i r s t , t h e l i n e s of the second; and the l i n e s of t h e second the points of t h e f i r s t . The above follows by noting t h e d u a l i t y of A l and A 1 l , t h e duality of A2 and A3, and t h e self-dual nature of A4. For convenience we may introduce t h e ordinary.geanetric language. Thus i f a point i s incident with a l i n e we say t h a t the point l i e s on t h e l i n e , or i s contained i n t h e l i n e , and t h e l i n e passes through the point. I f two points a r e incident on a l i n e we speak of the l i n e a s joining t h e tvo points.

I f a point i s incident with each of two l i n e s , ve say t h a t

the l i n e s i n t e r s e c t i n t h a t point.

With t h i s language ~4 may be re-

phrased as: A4.

Through any point P not lying on a l i n e Q , there pass exactly t

l i n e s intersecting E.

4. Graph of a p a r t i a l geometry. ( r , k, t ) i s defined a s follows. p a r t i a l geometry.

The graph G of a p a r t i a l geanetry

The v e r t i c e s of G a r e t h e points of the

Two vertices of G a r e joined (adjacent) i f t h e corres-

ponding points of t h e geometry are joined (incident with t h e same l i n e ) . Two vertices of G a r e unjoined (non-adjacent) i f the corresponding points

of the p a r t i a l geometry a r e unjoined ( i . e . there e x i s t s no l i n e incident with both t h e points 1. Theorem (4.4.1).

The graph of p a r t i a l geanetry ( r , k, t) is strong-

l y regular with parameters

lstzr,

lstsk.

Let t h e r e be v points and b l i n e s i n t h e p a r t i a l geometry.

Since

t h e points of t h e geometry have been i d e n t i f i e d with t h e v e r t i c e s of the graph G, we can c a l l two points of t h e geometry f i r s t associates i f they a r e joined by a l i n e , and second associates i f they a r e not joined by a Now through any point P of t h e geometry t h e r e pass r l i n e s , each

line.

of which contains k - 1 other points besides P. r(k

- 1 ) f i r s t associates.

Hence

This shows t h a t G i s a regular graph. ing through P. a s s o c i a t e s of P.

Hence P has exactly

Consider t h e b - r l i n e s not pass-

From ~4 each of these l i n e s contains exactly t f i r s t Any p a r t i c u l a r f i r s t associate Q of P, l i e s on r - 1

such l i n e s , since one of t h e r l i n e s passing through Q joins it t o P. Hence t h e number of f i r s t associates i s

By similar arguments we can prove [ ~ o s e(1963 b)] t h a t

Comparing (4.4.1) and (4.4.2) we have .,

(4.4.5)

b = r[(r-l)(k-l)+tYt,

and s u b s t i t u t i n g f o r b i n (4.4.3) we have

R . C . BOSE (4.4.6)

.n2 = ( r - l ) ( k - l ) ( k - t ) / t

- . We have now v e r i f i e d t h a t G

is strongly regular.

The v a l u e of v i s

given by (4.1.1) and t h e o t h e r parameters a r e obtained from (4.1.3) and (

1 4).

Hence theparameters of t h e s t r o n g l y r e g u l a r graph ( o r t h e assoc-

i a t i o n scheme) corresponding t o a p a r t i a l geometry ( r , k , t ) a r e given by

This a s s o c i a t i o n scheme may be c a l l e d t h e geometric a s s o c i a t i o n scheme with c h a r a c t e r i s t i c s ( r , k, t ) . Corollary.

For a p a r t i a l geometry ( r , k , t ) t h e number v o f p o i n t s

i s given by (4.4.7),

and t h e number b of l i n e s i s given by (4.4.5).

I f t h e v p o i n t s of t h e p a r t i a l g e m e t r y ( r , k , t ) a r e t a k e n a s t h e treatments, and t h e k l i n e s a r e taken a s t h e blocks of a design then it i s c l e a r t h a t a p a r t i a l geometry ( r , k , t ) i s a PBIB design based on t h e a s s o c i a t i o n scheme with parameters (4.4.71,

(4.4.8) and (4.4.91, f o r

which t h e parameters of t h e second kind a r e (4.4.10)

r,k,A

1

= 1 , A

2

= O .

Bose and C l a t ~ r t h y(1955) considered two c l a s s PBIB d e s i g n s with r < k , A1 = 1, A 2 = 0.

From t h e r e s u l t s o f Chapter 111, paragraph 5, it

follows t h a t f o r such designs t h e matrix

P*

given by (3.5.5) has a zero

c h a r a c t e r i s t i c r o o t , i f we t a k e Al = 1, A 2 = 0 .

Hence

Using t h i s r e l a t i o n , and t h e i d e n t i t i e s (3.1.3), (3.2.21,

(3.2.31,

(3.2.11,

(3.2.4) t h e y showed t h a t t h e parameters of t h e design

must be given by (4.4.7),

(4.4.8),

(4.4.9).

This r a i s e s t h e i n t e r e s t i n g

question, whether t h e design i s a p a r t i a l gecuuetry. affirmative.

(3.1.41,

The answer i s i n t h e

Since t h e axioms Al, A2, A3 a r e w i d e n t l y s a t i s f i e d , it

only remains t o show t h a t Ah i s a l s o s a t i s f i e d . Let K be t h e st of k treatments contained i n a p a r t i c u l a r block, and let

TI be

t h e s e t of remaining v - k treatments.

ber of treatments i n

Let g ( x ) denote t h e num-

which have exactly x f i r s t a s s o c i a t e s i n K.

Then

easy counting arguments show t h a t

Hence

x, t h e average value of x,

is

-

x = Iw(x)/Ig(x) = t ,

and

Hence x must always have t h e value t.

This i s equivalent t o t h e axiom ~ 4 .

Hence s PBIB desing with r r e p l i c a t i o n s , block s i z e k, X I =

1, A2 = 0, i s

R. C. BOSE a p a r t i a l geometry ( r , k, t ) i f r < k . One may ask whether a p a r t i a l geometry ( r , k, t ) e x i s t s f o r all values of r , k, t. ity

q of

Now for the corresponding PBIB design t h e multiplic-

t h e c h a r a c t e r i s t i c root

el

of the incidence matrix NN'

is g+

en by ( 3 . 5 . ~ ) . Substituting f o r nl, n2, v and A ha (4.4-71, (4.4.81, (4.4.9) and (3.5.8) we have

Hence a necessary condition f o r t h e existence of a p a r t i a l geometry

( r , k, t ) i s t h a t t h e number a given by (4.4.15) i s a positive integer. 1

For example i f r = 3, t = l t h e n t h e only possible values of k a r e k = 2, 3, 5 and 11. The cases k = 2 , 3, 5 are possible, but a rather lengthy combinatorial argument [ ~ o s eand C l a t ~ r t h y (1955)] shows t h e case k

= 11t o be impossible.

CHAPTER

v

THE mAMENTAL CIIARACTERIZATION THEOREM 1. Weakly balanced designs and graphs.

graph of valence d.

Let G be a f i n i t e regular

Let A(x,y) denote t h e number of v e r t i c e s which a r e

simultaneouly adjacent t o two given d i s t i n c t v e r t i c e s x ana y. defined t o be edge regular of edge-degree x , y of adjacent vertices.

6 if

Then G i s

A (x,y) = 6 f o r every pair

We s h a l l consider here a p a r t i c u l a r class of

edge regular graphs. Consider a s e t of v objects or treatments arranged i n t o b blocks or

R. C. BOSE s e t s , such t h a t each block c o n t a i n s a t l e a s t two t r e a t m e n t s , and t h e t r e a t m e n t s i n a given block a r e all d i s t i n c t .

Two treatments 0 and 9 a r e

c a l l e d f i r s t a s s o c i a t e s i f t h e r e i s a block c o n t a i n i n g b o t h 0 and 9. Otherwise t h e y a r e second a s s o c i a t e s .

The arrangement i s c a l l e d a weakly

balanced design i f t h e following conditions hold (1)

Each treatment occurs i n r blocks,

(2)

Two d i s t i n c t t r e a t m e n t s do not occur i n more than one block,

( 3 ) Each treatment h a s e x a c t l y r ( k

(4)

- 1 ) first associates,

Given any two t r e a t m e n t s 0 and 9 which a r e f i r s t a s s o c i a t e s

t h e r e a r e e x a c t l y 6 t r e a t m e n t s which a r e first a s s o c i a t e s t o both 0 and $. The condition ( 3 ) means t h a t t h e sum of t h e s i z e s of t h e blocks i n which a given treatment 0 appears i s r k .

Hence k i s t h e average s i z e of

t h e blocks i n which a g i v e n treatment appears.

Since each block c o n t a i n s

a t l e a s t twu treatments k h 2 . For any treatment 0 t h e r e e x i s t s a t l e a s t one block B containing 0 f o r which t h e block s i z e kl;k.

Let 9 be any o t h e r treatment i n B.

Then

t h e 6 treatments which a r e by (4) f i r s t a s s o c i a t e s t o both 0 and 9 a r e made up of t h e k l - 2

t r e a t m e n t s i n B o t h e r than 0 and 9 , t o g e t h e r with

t r e a t m e n t s which occur simultaneouly i n a block containing 0 but not 9 , and a block containing $ and not 0. 6 = kl-2+n>,k-2.

I f t h e r e a r e n such treatments t h e n

Let

Then a aO. For any p a i r of t r e a t m e n t s 0 and 9, l e t A(e,@) be t h e number of t r e a t m e n t s which a r e f i r s t a s s o c i a t e s of both 0 and $. i f 0 and $ a r e f i r s t a s s o c i a t e s .

Then A(B,$) = 6

Let 8 be t h e upper bound A(0,$) f o r a l l

p a i r s 6, @ such that 8 and @ a r e second associates.

Then 8 i s a non-neg-

a t i v e integer. The weakly balanced design under consideration w i l l be s a i d t o have t h e parameters ( r , k, a , 6 ) .

Note t h a t a p a r t i a l gecmetry ( r , k , t ) i s

a weakly balanced design ( r , k, a , 8 ) with

t h e blocks and treatments of t h e design being t h e l i n e s and points of t h e p a r t i a l ~eometry. The graph of a weakly balance (WB) design i s defined t o be a graph whose v e r t i c e s correspond t o t h e treatments of a design ahd f o r which two v e r t i c e s a r e adjacent i f and only i f t h e corresponding treatments a r e f i r s t associates.

The graph of a WB-design i s called a WB-graph.

Clearly

t h e graph of a WB design with parameters ( r , k , a , 0) s a t i s f i e s t h e following conditions

( c1) G i s regular of valence d = r ( k

- 1),

(c2) G i s edge-regular with edge-degree 6 = k - 2 + a , (c3) A(X,Y)< 8, f o r a l l p a i r s of non-adjacent v e r t i c e s , x and y of G. Here

d, 6 , r a r e p o s i t i v e i n t e g e r s , B > 0 i s a non-negative integer and

k22, a>O. A f i n i t e graph G s a t i s f y i n g t h e above conditions w i l l be c a l l e d a

pseudo WB-graph.

A pseudo WB-graph may not necessarily be t h e graph of a

WB-design. We s h a l l show t h a t i f k i s s u f f i c i e n t l y l a r g e i d comparison t o r, a and 8, then a pseudo WB-graph w i l l be t h e graph of a WB-design. 2.

Definitions.

We define here some functions of t h e parameters

r, k, a, 6 which play an important part i n t h e investigations which f o l . low.

We shall denote as usual t h e cardinality of a s e t S by

IS^.

A clique K of a graph i s a s e t of vertices adjacent t o each other.

A clique K w i l l be called complete i f we cannot f i n d a vertex x, not con-

tained i n K such t h a t xUK i s a clique.

Thus a complete clique cannot be

extended t o a larger clique by t h e adjunction of a new vertex belonging t o t h e graph. Now consider t h e graph G with t h e properites, (cl), (c2), (c3).

A

clique K of G w i l l be called a major clique i f (5.2.5)

l~l;l+k-y(r,a)

= k-(r-1)a.

The clique K of G w i l l be called a grand clique i f it i s bo$h major and complete. A claw b , S ] of G, consists of a vertex p, t h e vertex of the claw,

and a non-empty s e t S of v e r t i c e s of G, not containing p, such that p is adjacent t o every vertex i n S, but any two v e r t i c e s i n S a r e non-sdjacent. The order of t h e claw is defined t o be t h e number s =

. 3.

Theorems and lemmas f o r claws i n pseudo WB-graphs.

pseudo -graph

Let G be a

satisfying conditions (5.1.3).

Theorem (5.3.1) r + l i n G.

IsI.

If k > p ( r , a , $ ) , there cannot e x i s t a claw of order

R. C. BOSE Suppose there exists i n G a claw Cp,$1 of order s. of vertices of G , not belonging t o

Cp,sI, and

Let T be the set

aqlacent t o p.

Let f ( x )

denote the number of vertices q in T, such that q i s adjacent t o exactly Counting the number of vertices i n T we have from (cl),

x vertices i n S.

Counting the number of ordered pairs (b,q) where b

asid

q are adjacent,

b belongs t o S, and q belongs t o T, we have from (c2)

Again counting the t r i p l e t s (bl, b2,

Q),

where bl, b2 i s an ordered

pair of vertices i n S, q i s a vertex i n T, and bl b2, are both adjacent t o q, we have from (c ) 3

I f a claw of order r + l exists, putting s = r + l , we have from (5.3,1),

(5.3.2)

and (5.3.3):

Since the left-hand side i s essentially non-negative whereas k > p(r,a,b) by hypothesis, we have a contradiction.

This proves our

theorem.

The following Lemmas are readily prwed: Lemma (5.3.1). be extended t o

8

If k > y ( r , a ) , then any clawof G of order s < r , can

claw of order r.

R . C. BOSE Given a claw

Lemma (5.3.2). least k

- y( r,a)

Cp,s] of

G of order r -1 t h e r e e x i s t a t

d i s t i n c t v e r t i c e s q of G such t h a t [p,SUq] i s a claw of

order r. Theorems and lenrmss f o r cliques i n pseudo WB-graphs.

4.

l e t G be a graph satisf'ying t h e conditions (el),

( c 2 ),

(C

3

As before

).

I f k>max[y(r,a), p ( r , a , ~ ) ] , then any p a i r of adja-

Lemma (5.4.1).

cent v e r t i c e s p and q i s contained i n a t l e a s t on grand clique.

From Lemma (5.3.1) we can extend the claw [p,q] t o a claw order r.

Let bl, b2,

..., br be v e r t i c e s

i n S other than q.

[ p , ~ofl

Let 0 be

t h e s e t of v e r t i c e s w, which when adjoined t o S - q give a claw [ p , ~ *1of

-

order r , where S* = IS q ) ~ o . Of course q i s contained i n O and from

Lemma (5.3.2)

The v e r t i c e s i n O are sll adjacent t o one another. not adjacent they could be added t o bl, b2, order r

+ 1, which

I f any two were

..., br t o give a claw of

would contradict Theorem (5.3.1).

Let K = pU0.

Then K

i s a major clique since

We can extend t h e major clique K by adding new v e r t i c e s till it i s complete and therefore a grand clique. The following Lemmas a r e e a s i l y proved: Lemma (5.4.2). c l i q u e , then

I f K and L a r e cliques of G, and KUL i s not a

I K ~ L I 2 6.,' .

Lemma (5.4.3).

I f K and L a r e cliques of G and K

l e a s t two v e r t i c e s a and b, then / K U Lemmas (5.4.4).

LI

f

L contains a t

k + a.

I f K and L a r e cliques of G, K cl L i s not a clique

R. C . BOSE and K f l L contains a t l e a s t two v e r t i c e s , then

Theorem (5.4.1).

I f k>max[y(r,a), p(r,a,B), p(r,a,B)] then any

pair of adjacent v e r t i c e s p and q is contained i n one and only one grand clique. The existence of a t l e a s t one grand clique follows a t once from Lemma (5.4.1). Suppose t h e r e e x i s t a t l e a s t two d i s t i n c t grand cliques K and L both containing t h e adjacent vertices p and q. K U L i s not a clique.

Since K and L axe complete,

Hence from Lemma (5.4.4)

But K and L a r e both major cliques.

Hence

which shows t h a t

contrary t o t h e hypothesis. Theorem (5.4.2).

I f k>max[y(r,a), p(r,a,B), p ( r , a , d ) ] t h e n each

vertex of G i s contained i n exactly r grand cliques. From Theorem (5.4.1) any pair of adjacent v e r t i c e s i s contained i n exactly one grand clique. l e a s t one claw of order r. b2,

..., br).

Also from Lemma (5.3.1)~ p i s t h e vertex of a t Let[p, S] be a claw of order r , where S = Ebl,

A s i n Theorem (5.3.1), l e t T be the s e t of v e r t i c e s not

belonging t o S, which a r e adjacent t o p. Let H be t h e s e t consisting of p, bj, and q belonging t o T , such j

t h a t q i s adjacent t o b but not adjacent t o bi, i # j. As i n Theorem j

R. C . BOSE (5.3.1) l e t f ( x ) denote t h e number of vertices i n T which are adjacent t o exactly x vertices i n S. Then f ( 0 ) = 0, otherwise there would e x i s t a claw of order r + l .

Putting s = r i n (5.3.1) and (5.3.2), we have

Any two vertices of

H a r e adjacent t o one another, otherwise t h e r e

5 w u l d e x i s t a claw of order r + l . Put H* = H5

5

- (b5 U p).

T which a r e adjacent t o b

..., H:

5

Thus H i s a clique. 5 Then H* consists of exactly those v e r t i c e s of

5

but t o no other vertex of S.

Hence HT, H;,

a r e d i s j o i n t s e t s , and t h e t o t a l number of v e r t i c e s i n these s e t s

Now there i s a unique grand clique K containing b and p.

5

5

The num-

ber of v e r t i c e s i n K cannot be l e s s than the number of vertices i n H 5 3' I f possible l e t I < I H ~ Since K i s a grand clique it follows t h a t 3 'I H i s a major clique and contained i n scme grand clique K' Since b

IK

1.

5

5'

j

and p a r e contained i n K and K' theXmust coincide. Hence K contains 3 5' 3 H which contradicts IK I < I. 5' 3 Now consider t h e r grand cliques, K2, Kr. Then K1-p,

IH~

5,

K2-p,

..., Kr-p

are d i s j o i n t .

...,

For i f Ki-p and K - p , i # 5 , have a

5

common vertex q, then Ki and K would coincide, and would contain both

5

bi and b

j'

which i s impossible since bi i s not adjacent t o b

ing (5.4.3) we can now deduce

j'

Remember-

R.

C. BOSE

I f possible, suppose there is another grand clique K*l p.

containing

The vertices in Kr+l-p must be disjoint from the vertices i n K1-p,

K2-p,

..., Kr-p.

Since KHl

i s a grand and therefore a msjor CZfque,

I K - p~ 1 3 ~k - 1- (T - 1 ) s .

But from (el), thr number of vertices sdjacent

t o p i s exactly r ( k - 1 ) .

Hence fram (5.4.4)

which i s a contrdiction.

Thus p i s contained i n exactly r grand cliques.

5.

The fUndamental characterization theclrem.

Theorem (5.5.1). (c2), (c3).

Let G be a graph sati6fying the conditions (cl),

Then if

i s the graph of a WB-des-

with parameters ( r , k, a , 8).

I f we take t h e vertices of G t o be the treatments and the grand cliques of G t o be t h e lines of the design then it follows directly from the theorems proved &n the l a s t two paragraphs that the design is a weakl y balanced design with parameters ( r , k, a, 6).

The above

$8

a slightly different version of the corresponding re-

milts i n Bose and Lasker (19671, which mey be consulted for *ther t a i l s of the proof.

de-

R. C . CHAPTP(

BOSE

VI

SOME SPECIAL GRAPHS 1. Pseudo uearmetric maphe.

A strongly regular graph i s defined t o

1 be pseudo geometric ( r , k, t ) i f i t s parameters v, nl, pll, given by (4.4.7) and (4.4.4) where

l z t s r , 1st sk. -

2 pU a r e

~ h u sa pseudo

geometric graph ( r , k, t ) has t h e same parameters as the graph of a par-

.

tial geometry ( r , k, t ) [Cf paragraph 4, Chapter IV].

However, a pseudo

geometric graph may not be t h e graph of a p a r t i a l geometry.

If G i s the

graph of a partial geometry ( r , k, t ), G may be called a geometric graph (r, k, t ) .

We shall e s t a b l i s h a sufficient condition f o r a pseudo geo-

metric graph ( r , k, t ) t o be geometric ( r , k, t ) . I f G i s a pseudo geometric graph ( r , k, t ), then it i s a pseudo WBgraph with parameters ( r , k, a, 6) , [ ~ f .Chapter V, paragraph I]', where

In t h i s case t h e functions y, q, 0 , p given by (5.2.1) become

Note t h a t i n view of t h e inequality 1lt:r

and

we have

- (5.2.4)

R. C. BOSE It follows from the fundamental characterization Theorem (5.5.1)

that a pseudo geometric graph ( r , k, t ) i s the graph of a WB-design D with parameters ( r , k, a , 8 ) i f k >/ p(r , t ), where a and 6 a r e given by (6.1.1). I f we take t h e treatments of the design as points, the blocks of the designs as l i n e s , and incidence as the relation of a treatment being contained i n t h e design, then fran t h e definition of a WB-deaign the axioms

Al and A2 f o r a partialrgeoemtry are satisfied.

We shall show that axi-

oms A3 and ~4 are also satisfied. Let 8 be any treatment of D and l e t B1, containing 8. block B

3

B2,

..., Br

be the blocks

Since the average s i z e of these blocks i s k, the largest

( 1 6 j \< r ) contains a t l e a s t k treatments.

Let B be a subset of

k treatments contained i n B Let be t h e s e t of treatments not con,I' tained i n B. Let g(x) be the number of treatments i n which have exactly

x

f i r s t associates i n B.

Then an easy counting argument shoys that

and

Hence the average value of x i s t and

which i s only possible i f x i s constant and equal t o t. Hence every treatment i n

'ii has exactly t f i r s t associates i n

If B contains any treatment j

B, then O belongs t o

5 and

B.

+ other than those already contained

in,

therefore, has exactly 't f i r s t associates i n B.

R. C. BOSE But each treatment of B i s a f i r s t associate of O .

t = k > p ( r , t ) > r , which contradicts block B1, Be,

..., Br

i s of s i z e k.

axiom A3 i s satisfied.

ter.

Hence

This shows that each of t h e

Since 8 i s any arbitrary treatment

AlaO x = t means that Ab is a l s o satisfied.

Hence t h e design D i s a p a r t i a l gecm~etry. We therefore-have t h e following theorem [ ~ o s e(1963 c ) 1 Theorem (6.1.1).

I f t h e graph G is pseudo grametric ( r , k, t ) then

it is geometric i f

2.

Triangular and pseudo triangular graphs.

The triangular assoc-

i a t i o n scheme with parameters given by (3.3.3) and (3.3.4) may be denoted by T2(m).

The corresponding strongly regular graph with paramenters

(6.2.1)

v = m(m-1)/2,

n = 2(m-2). 1

1 pll = m-2,

will also be called a triangular or a T2(m) graph.

:p

=4

The m(m

9

- 1)/2 t r e a t -

ments of a T2(m) association scheme may be made t o correspond t o t h e m(m-1)/2 unordered p a i r s chosen out of a s e t of m symbols 1, 2,

..., m.

The p a i r ( i , j ) corresponds t o t h e treatment occuring i n t h e i-th row and j-th column of t h e m x m square i n which t h e treatments a r e exhibited. Thus i n t h e example m = 5, given i n paragraph 3(b) of Chapter 111, t h e treatments 5, 8,10 corresponds t o t h e pairs (2,3), (3,4) and (4,5) respectively. i.e.,

It i s c l e a r t h a t i f two treatments axe f i r s t associates,

a r e i n t h e same row or column, then t h e corresponding p a i r s have a

common symbol and i f they a r e second associates, i.e.,

do not occur i n

t h e same row or column, then t h e corresponding p a i r s have no symbol i n common.

I n t h e example under consideration, t h e treatments 5 and 8 cor-

responding t o t h e p a i r s '(2,3) and (3,4) a r e f i r s t associates, and t h e

R.

C . BOSE

treatments 5 and 10 corresponding t o the pairs (2,3) and (4,5) are sec-' ond associates.

Hence we'can l a b e l the vertices of a T (m) graph G by 2 the m(m -1)/2 unordered pairs of symbols selected *om 1, 2, , m.

...

Then two vertices are adjacent or non-adjacent according aa t h e c a r r e sponding p a i r s have or do not have a camon symbol. A strongly regular graph G with the parameters (6.2.1)

One may ask whether a pseudo

pseudo triangular or pseudo T2(m) graph. triangular graph i s triangular, i .e. by the unordered pairs ( i , j ),

, can

w i l l be called

-

the m(m 1 ) vertices be labeled

15 i, j 5 m such t h a t two vertices are ad-

jacent i f and only i f the corresponding pairs b e a common symbol. answer is i n the affirmative i f m # 8.

The

I f m = 8 then besides the graph

T2(8) there a r e three other non-isomorphic -graphs. Now comparing (6.2.1)

with (4.4.7) and (4.4.4) it i s clear t h a t a

pseudo triangular graph ~ ~ ( m is )pseudo gecmetric (2, m - 1 , (6.1.5), p(2,2) = 7. geometric (2, m '

2).

From

Hence from Theorem (6.1.1) a pseudo 'P2(m) g r e h i s

- 1, 1 ) provided m > 8.

Thus a pseudo triangular graph T2(m), i s the graph of a p a r t i a l geo-

metry (2, m

- 1, 2).

Let t h i s graph be G.

Frorm the corollary t o Theorem

(4.4.1), the number of l i n e s i n t h e geametry i s m, and t h e number of points is m(m-1)/2. point.

Since r = t = 2, any two lines intersect i n a unique

We may therefore label t h e l i n e s by the symbols 1, 2,

...,m;

and

the point of intersection of the l i n e i and the l i n e j , i # j may be labeled by the unordered pair ( i , j ) .

It i s now clear t h a t two points are

incident with the same l i n e i f and only i f the corresponding pairs have a common symbol.

This shows that G i s a triangular graph.:

We have thus shown that a pseudo triangular graph ~ ~ ( r ni s) triangular if r n > 8.

This result i s due t o Connor (1958).

The same result

R. C. BOSE was proved by Shrikhand~(1959 a ) f o r m; 6,- and by ~ o i f b ~ e(1960 n a) for t h e case m = 7.

However, it is surprising t h a t when m = 8, t h e parameters It turns aut t h a t there a r e

(6.2.1) do not characterize a T2(8) graph. three other non-isanorphic graphs. (1960 b ) and by Chaw

C (19591,

This was damnatrated by Ho-

(1960) I.

Hoffman's proofs use t h e f a c t t h a t i f G is a strongly zcgular graph with parameters given by (6.2.1),

then the characteristic roots of its

adjacency matrix are

-

(a)- 2m 4 with multiplicity 1, and characteristic vector (1, 1,

...,

1). (b) m (c)

- 4 with multiplicity m - 1.

-2 with multiplicity v-m.

Shrikhande's and Chang's proofs use purely ccanbinatorial arguments. W e can summarize these r e s u l t s i n the following theorem. Theorem (6.2.1).

I f G is a pseudo triangular graph with parameters

given by (6.2.1), then it is triangular i f m

# 8. When m = 8 then t h k

are three other non-isomorphic graphs besides T2(8).

It i s of interest t o exemine the nature of t h e other three pseudo triangular graphs ~ ~ ( with 8 ) parameters v = 28,

nl = 12,

1 pU = 6,

2 pll = 4

.

The condition (b) of Theorem (4.2.2) i s s a t i s f i e d since

-

W e s t a r t with the graph ~ ~ ( 8 whose 1 , vertices a r e t h e unordered p a i r s

which can be formed frcm t h e symbols 1, 2,

..., 8 and i n which two ver-

t i c e s a r e adjacent i f and only i f the corresponding pairs have a comon symbol.

Theorem (4.2.2) now shows that i f we obtain ~ ~ ~from ( 8~ )~ ( by 8 )

R. C.

BOSE

complimentation with respect to V1, where V is any of the three sets, 1 and V is the set of vertices of T2(8), and V2 = V-Vl; .S1, S2 or S3 then T;*(8) has the same psramsters as ~ ~ ( 8 ) .The sets S1, S2, S -are 3 given below S, = {(12),(34),(56),(78)1, S2 = {(12),(34),(56),(78),(13),(24),(57),(68)}, S3 = {(12),(34),(56),(78),(13),(24),(57),(68),(14)~(23),(58),(67)}~ ~ ( cannot 8 ) have any claw of order 3, and there is no complete quadrangle.

The non-isomorphism of the other three graphs can be checked

by counting claws of order 3, and the number of complete quadrangles. 3.

The Hall-Connor embedding theorem.

sign with parameters v = b, r = k, A.

Consider a symmetric BIB de-

We have shown in Theorem (1.5.1)

that we can obtain from it another BIB design D*, the residual of D by deleting one block and all the treatments contained in this block.

The

parameters of D* are

One may ask the following question: meters (6.3.1),

Given a BIB design D* with para-

is it possible to obtain from it a symmetric BIB design

with parameters v = b, r = k, X by taking an additional block of k new treatments and adjoing to each of the blocks of D*, X suitably chosen treatments out of the k new treatments? When A = 1, D* is isomorphic to an affine plane, and can be converted to a projective plane (isomorphic to D) in the usual manner. A

Hence for

= 1 the result is certainly true. Hall and C o m r (1953) showed that

the result is also true for A = 2. was given by Bhaltacharye (1944)

.

For the case A = 3 a counter example

R. C. BOSE Shrikhande (1960) gave an alternative proof of t h e He.ll-Connor theorem depending on the characterization theorem (6.2.1) for pseudo tri-

angular graphs. For the case X = 2, t h e parameters of the residual design Dff can be written a s

, It i s readily proved t h a t two blocks two treatments.

of D intersect i n either one or

I f we take the blocks of D. a s the treatments of an as-

sociation scheme and c a l l two blocks f i r s t associates or second associates according a s they intersect i n one treatment or two treatments then it can be shown [ H a l l and Conqor (1953), Shrikhande (1952) 1that we get a two class association scheme with parameters

Hence the corresponding strongly regular c a p h i s the pseudo triangular graph ~ ~ ( kwhich ) , by Theorem (6.2.1)

k # 8.

If we now take k symbols 1, 2,

is triangular T2(k) i f

..., k then t o each block of Dff we

can assign an unordered pair ( i j) of symbols such t h a t two blocks are f i r s t associates, i.e. intersect i n one treatment i f and only i f the corresponding pairs have one symbol i n comon. ments.

We can now take k new t r e a t -

I f (ij ) i s the pair assigned t o the block B

id

adding two new treatments i and j. of all the newtreatments.

we extend it by

Finally, we add a new block consisting

We have now obtained a design with

-

2 (k k + 2112 blocks and treatments, with each treatment appearing i n k blocks of s i z e k, such t h a t any two blocks intersect i n two treatments.

From t h i s it follows t h a t every pair occurs i n two blocks.

This proves

R. C. BOSE Hall and Connors theorem i f k # 8.

Connor (1952) gave a separate proof

f o r the non-existence of (6.'3.1) f o r t h e case k = 8.

Hence t h e problem

of embedding does not a r i s e .

4.

Enbedding t h e complement of an oval i n a projective plane sP even Given a projective plane of even order q, it i s w e l l known hose

order.

(1947), Segre (1954) and (1955)l t h a t we can find a s e t of q + 2 points i n II such t h a t no t h r e e a r e collinear.

They form an oval i n

n.

I f we delete

the points of t h e oval from II, then t h e incidence structure of t h e remaining points and l i n e s has t h e following properties: (a)

-

There a r e q2 1 points.

(b) There a r e two types of l i n e s . with q + l points.

Lines of type I a r e each incident

Lines of type I1 a r e each incident with q - 1 points.

( c ) Each point i s incident with q/2 l i n e s of type I and ( q + 2112 points of type 11. (d) Any two d i s t i n c t points a r e both incident with exactly one l i n e which may be of type I or type 11. Conversely l e t 0 be an incidence structure with the above properties. We may ask t h e question whether it i s possible t o embed 0 i n a projective planeoforder q by suitably extending t h e l i n e s of type 11. Bose and Shrikande (1972) have proved t h a t t h e answer i s i n the affirmative except possibly f o r t h e case q = 6.

Their proof essentially depends on showing

t h a t i f we take t h e l i n e s of type I1 a s t h e treatments of an association scheme and c a l l t h e two l i n e s f i r s t associates i f they do not i n t e r s e c t and second associates i f they i n t e r s e c t

i n a point, then t h e association

scheme has t h e parameters (6.2.1) with m = q + 2.

Hence from Theorem

(6.2.1) t h e corresponding graph i s triangular except possibly i n the case q =

6.

This enables them t o suitably extend the l i n e s .

R. C.

5. Net and pseudo n e t graphs.

BOSE

A net (r, k ) of degree r and order k

i s a system of undefined points and l i n e s together with an incidence rel a t i o n subject t o t h e following axioms ( i ) There i s a t l e a s t one point ( i i ) The l i n e s of t h e net can be partitioned i n t o r d i s j o i n t , nonempty, "parallel classes" such t h a t each point of t h e n e t i s incident with exact-

tw l i n e s belonging t o d i s t i n c t class-

ly one l i n e of each c l a s s and given

e s t h e r e i s exactly one point of t h e net which i s incident with both l i n e s ( i i i ) One l i n e i s incident with k points. For convenience we can use phrases such a s "point i s on a l i n e " instead of speaking incidence.

Then it can be r e a d i l y proved

C B ~ U C ~(196311

that (1)

Each l i n e of t h e net contains exactly k d i s t i n c t points where

-

k>l.

(2) Each point of t h e net l i e s on exactly r d i s t i n c t l i n e s where

ril. (3)

The net has exactly r k d i s t i n c t l i n e s .

p a r a l l e l c l a s s e s of k l i n e s each. c l a s s have no common points.

These l i n e s f a l l i n t o r

Distinct l i n e s of t h e same p a r a l l e l

Two l i n e s of d i f f e r e n t c l a s s e s have one

common point.

(4)

The net has exactly k

2

d i s t i n c t points.

It i s easy t o show b s e (1963 c ) 1t h e equivalence of a n e t ( r , k ) , a p a r t i a l geometry ( r , k , r

- 1 ) and

a s e t of r

- 2 mutually orthogonal

Latin squares of order k. The graph of a net ( r , k) o r t h e corresponding p a r t i a l geometry ( r , k, r

- 1) can be defined.' a s usual.

The points of t h e net o r t h e p a r t i a l

geometry correspond t o t h e v e r t i c e s of t h e graph and t m v e r t i t e s are adjacent i f and only i f t h e corresponding points a r e incident with t h e same

H. C. BOSE For r 2 3 very l i t t l e i s known a s t o what happens when k s p ( r , r - 1 ) . i s an open question whether p(r,r

It

- 1 ) i s the best possible value f o r the re-

s u l t t o hold, i.e. does there dways e x i s t a graph non-isoororphic t o the net graph ~ , ( k ) when k = p(r,r 6.

The B r u c k - & M e

-1 ) .

.

ambeddim theorem.

Bruck defines the def i-

ciency d of a net ( r , k) by d = k-r + l

(6.6.1)

.

The interpretation of the dZficiency d i s that i f it were possible t o add d more p a r d l e l classes, each consisting of k lines, so t h a t the extended net now has k + l classes of parallels, the net would become an affine plane, i n which any two points are joined by a unique l i n e .

W e

shall now prove: Theorem (6.6.1).

A net ( r , kl of 'deficiency d can be completed t o

an affine plane of order k i f

From the equivalence of s s e t of r - 2 mutually orthogonal Latin squares of order k and a net ( r , k) we can write the above theorem i n the following equivalent form Theorem (6.6.2

r.

If there exist k

- 1- d mutually orthogonal

Latin

squares of order k, it i s possible t o get a complete set of k - l m u t u a l l y orthogonal Latin squares, by adding d new suitably chosen squares, provib 1

ed t h a t k ~ ~ ( d - l ) ( d ~ - d ~ + d + ~ ) . The case d = 2, was f i r s t obtained by Shrikhande (1961) and the gene r a l case was obtained by Bruck (1963).

When d

s

2, the theorem holds

for k >4, but k = 4 is exceptional a s it i s well known t h a t a cyclic

Latin square of order 4 cannot belong t o a complete set of three mutually

It follows fram !ChArem (4.4.1) and can e a s i l y be v e r i f i e d direct-

line.

l y that t h e parameters of t h e graph of a net ( r , k ) s r e (6.5.1) v

= k2 , nl = r ( k - 1 1 ,

pll

2 = r(r-1) pu

= ( r - r - 1 - 2 ,

The graph of a net ( r , k) w i l l be called a net graph Lr(k). with parementers (6.5.1)

wiu

be called a pseudo net graph Lr(k).

pseudo net graph Lr(k) i s not necessarily a net graph. consequence of Theorem ( 6.1.1) Theoren (6.5.1).

A graph

, we

A

However, as a

have

A pseudo net graph Lr(k) i s a net graph Lr(k) i f

k>p(r,r-l) = $r-l)(r3-r2+r+2)

.

The special case r = 2 was f i r s t proved by Shrikhande (1959 b )

.

In

this case i f k > 4, then a pseudo net graph L2(k) i s a net graph L2(k).

It i s easy t o check t h a t t h i s r e s u l t i s also t r u e f o r k < 4.

But t h e case

1; = 4 i s exceptional. For a net (2, k) a point P may be given coordinates ( i , j ) , 1 4 i,j ,< k i f it i s on t h e i-th l i n e of t h e f i r s t p a r a l l e l c l a s s and j-th l i n e of t h e second p a r a l l e l class.

Then v e r t i c e s of t h e -net graph L2(k)

a r e t h e ordered pairs ( i , j ) two v e r t i c e s being adjacent i f sad only i f they have a common coordinqte. s a t i s f i e d i f k = 4.

The condition (b) of Theorem (4.2.2) i s

The condition ( a ) i s also s a t i s f i e d i f we take

The 'graph G* obtained by complementation with respect t o V and V 1

2

has

t h e parameters (6.5.1) f o r k = 2, but i s non-ismorphic t o the net graph L2(4). The general case of Theorem (6.5.1) was obtained 'by Bruck (1963).

orthogonal Latin squares. The proof of these tu6 theorems depends on T h m (6.5.1). be a pseudo net graph L, (k)

.

let G

Then its parameters are givcn by (6.5.1). i

The other parameters n

and pjk ( i , J , k = 1, 2) are given by (3.342),

(3.3.13) and (3.3.14).

Let

2

of

5 are

5 be

t h e canplementary of G, i.e. the vertices

the same a s the vertices of G but the vertices adjacent in G are

non-adjacent i n

and vise-versa.

-a

Then the parameters of G.cue obtsined

from G by reversing the subscripts 1 and 2.

Hence k2,

;=

Thus

i s strongly regular with parameters

El =

d(k-1).

-1

p

d

-

1

-

is pseudo ~ ~ ( k )From . Theorem (6.5.1)

Thus

i f (6.6.2) is satisfied.

2

-

2

%-d(d-1)

it i s the net graph Ld(k)

Thus there e x i s t s a net (d, k) with the arrc

points a s the net ( r , k ) but i f twu points are incident o r non-incident with the same l i n e i n ( r , k), they they a r e mn-incident o r incident v i t h the same l i n e i n (d, k).

Adding the d k l i n e s of the net (d, k) t o the

( r , k) l i n e s of the net ( r , k) we have a net (k + 1, k ) vhich is plane of order k.

7. 1, 2,

This proves the required reeult. Consider a s e t S of m symbols

T (m) and pseudo T (m) graphs.

..., m where m t 2.

elements from S, 1 6 q r m. dered q-plets of S.

8ffi~e

We can form ()'

Q

unordered q-plets of q d i s t i n c t

Let G be a graph wtrose vertices are the uuor-

Let two vertices of G be adjacent i f the correspond-

ing q-plets have exactly q - 1 symbols of S i n cammn, and nonadjacent

R.

otherwise.

We s h a l l c a l l G a T (m) graph. '4

C. BOSE

Clearly G has t h e follo&

properties: (1)

G i s regular of valence d

-

= q(m q).

( 2 ) G i s edge regular with edge degree 6 = m

- 2.

( 3 ) I f x and y a r e non-adjacent v e r t i c e s of G then A(=, y ) 2 4. A graph v i t h (m) v e r t i c e s , and satisfying t h e conditions ( I ) , ( 2 ) q

and (3) w i l l be called pseudo T (m) graph. We can then ask under what P conditions a pseudo T (m) graph is a T (m) graph. 9 Now t h e conditions (11, (2) and ( 3 ) a r e t h e same a s t h e conditions (cl),

(c2) and (c3) of Chapter V, paragraph 1, provided we s e t

(6.7.1)

r = q,

k = m-q+l,

a = q-1,

6= 4

.

Also d, 6 , r a r e p o s i t i v e integers, 0 i s a non-negative integer and k > 2, a > 0. Hence a pseudo T (m) graph i s a pseudo WB-graph. P The functions y ( r , a ) , q ( r , a ) , p(r,a,B);p(r,a,B) given by (5.2.1)

-

(5,2.4) now become

Clearly maxCy(r,a), ~ ( r , a , B ) , p(r,a,B)I = 4 + ( 2 q - l ) ( q - 1 ) . Hence from t h e fundamental characterization theorem a pseudo T (m) q graph G i s t h e graph of a WE-design with parameters ( r , k, a , 0 ) given by (6.7.1), (6.7.6)

i f k> 4 + (2q-lnq-11,

i.e.

rn>4+2q(q-l)

.

The treatments of t h e WB-design correspond t o t h e v e r t i c e s of G and

R. C . BOSE

theblocks of t h e WB-design'correspond t o t h e grand cliques of G.

Bose

and Lasker (1967). showed for t h e case m = 3 and Dowling (1969) f o r t h e g e n e r d case t h a t i f we take t h e grand cliques of G as t h e v e r t i c e s of a new graph G* and consider two v e r t i c e s of G* adjacent i f t h e corresponding grand cliques both contain a vertex of G i n canon then G" s a t i s f i e s the conditions (I), (2) and ( 3 ) if we replace m by m* = m

- 1.

Using in-

duction we have Theorem (6.7.1).

If for a graph G with (m) vertices t h e conditions '4 (11, ( 2 ) and (3) a r e s a t i s f i e d , i.e. G i s a pseudo T (m), then G i s a P T (m) graph i f 9

Notice t h a t t h e case q = 2 reduces t o Connor's r e s u l t t h a t f o r m > 8 a pseudo triangular graph T2(m) i s triangular T2(m).

In particular i f a graph G with

m(m

- l)(m - 2)/6 v e r t i c e s s a t i s f i e s

t h e conditions (11, ( 2 ) and (3) with q = 3 then G i s a ~ ~ ( graph m ) if

m >16. Aigner (1969) haa shown t h a t t h e same hold i f m c 9. t i o n i s open f o r 9 c m

c 16 though

m e ques-

no exceptional cases are known.

For

q > 3 nothing i s known about t h e case when m \ < 2 q ( q - I ) + & .

8.

Characterization of cubic l a t t i c e maphs.

A cubic l a t t i c e graph

of order m i s a graph G whose v e r t i c e s can be identified with t h e ordered t r i p l e t s on m symbols so t h a t txo v e r t i c e s a r e adjacent i f t h e corresponding t r i p l e t s have conrmon symbolt3 i n exactly two positions. 3

If G i s

a cubic l a t t i c e graph of order m then G has m vertices and possesses the following properties. (i) (ii)

G i s regular with valence d = 3(m

- 1).

G i s edge regular w i t h edge degree 6 = m

- 2.

R. C . BOSE (iii)

I f x and y a r e non-adjacent vertices of G, then A(x, y) = 2,

i f d(x, y ) = 2, and ~ ( x y , ) = 0 i f d(x, y).>2, where d(x, y ) denotes t h e distance between x and y. Lasker (1967) and Dowling (1968) have proved Theorem (6.8.1).

A graph G with m

3

vertices and possessing the pro-

p e r t i e s ( i ) , ( i l ) and ( i i i ) i s a cubic l a t t i c e graph of order m i f m > 7. Lasker's original characterization had an additional assumption which The s t a r t i n g point of t h e i r proofs i s t h a t the

vas eliminated by Dowling. p p e r t i e s (i), (

i

a n d . ( i i i ) imply the p r ~ p e r t i e s(cl), (c2) and (cl)

of Chapter V, paragraph 1 i f

Hence

Therefore the fundamental characterization theorem (5.5.1) applies

CXAPTm VII

GRAPHS IN WICH EACH PAIR OF VERTICES IS ADJACENT TO THE SAME NUMBER d OF OTHER VERTICES 1

1. Graphsfor which p,

and

graph not necessarily connected.

$ a r e constant.

L e t G be a f i n i t e

If x and y are any tw vertices l e t

~ ( x y, ) denote the number of vertices sirmiLtaneous1.y adjacent t o both x

R . C . BOSE

and y.

Bose and Dowling (1971) proved I f f o r a f i n i t e graph G, A(x, y ) = pll1 i f x and y

Theorem (7.1.1).

a r e adjacent, and A(x, Y ) = p 2 and pll

2

ll

i f x and y are not adjacent, where p1 11

a r e f i x e d non-negative i n t e g e r s , then only t h e following cases

a r e possible (i)

G i s regular and hence strongly regular.

( i i ) pl: 1

p11+2,

= 1, and G c o n s i s t s of n complete subgraphs of order

with one common vertex.

(iii) 1 order p11+2,

p& = 0, and G consists of n d i s j o i n t complete subgraphs of and m i s o l a t e d points.

The proof depends on simple counting arguments and a number of Lemmas on t h e existence of 3 and 4 cycles i n the graphs. 2.

G ( d ) graphs and symmetric BIB designs.

A f i n i t e graph G i s de-

2

t o be a G ( d ) graph if any two d i s t i n c t v e r t i c e s x, y a r e both ad2 jacent t o exactly d other v e r t i c e s . Hence A(x, y ) = d whether x and y are 1 = d. Thus Theorem (7.1.1) applies with pll = adjacent o r non-adjacent.

fined

Thus a ~ ~ ( graph d ) i s necessarily regular i f d

3

2.

Again from the same

theorem we have Lemma (7.2.1).

An i r r e g u l a r ~ ~ ( graph d ) with d = 1, must consist of

n > , 2 complete graphs of order 3 with a common vertex. The case d = 0 i s t r i v i a l .

It i s c l e a r t h a t i n t h i s case t h e graph

consists of m i s o l a t e d v e r t i c e s , and n d i s j o i n t edges together with t h e i r 2n v e r t i c e s . We can t h e r e f o r e confine our a t t e n t i o n t o regular ~ ~ ( graphs. d )

A

regular G (d) graph with v v e r t i c e s and valence nl w i l l be s a i d t o have 2 pmameters ( v , nl, d ) .

We can obtain from t h i s a symmetric B I B design f o r

which r = k = n1, X = d i n the following manner:

Let t h e v e r t i c e s of t h e

graph be taken as treatments.

Let t h e i-th block consist of a l l t r e a t -

ments which correspond t o v e r t i c e s adjacent t o t h e i - t h vertex.

Then

c l e a r l y each block i s of s i z e nl and each treatment occurs i n nl blocks. Also any two treatments x and y w i l l both occur i n t h e block i i f and only i f t h e vertex i i s adjacent t o both t h e v e r t i c e s x and y. .any two treatments occur together i n d blocks.

ij

It follows from (1.1.1)

This BIB design has t h e property (S) t h a t i t s

t h a t v = b = n (n - l ) / d . 1 1 incidence matrix B = (n

Hence

) can be written i n a form such t h a t nij = nji

f o r i # j and nii = 0. A symmetric BIB design with v treatments, block s i z e k and f o r which every p a i r of treatments occur inA block w i l l be c a l l e d a symmetric (v, k, A ) B I B design.

It i s c l e a r t h a t from a symmet-

r i c (v, k, A) BIB design, with t h e property ( s ) , we can obtained a G ~ ( A ) graph with parameters Theorem (7.2.1).

(r, k,

A).

Hence we have

The existence of a ~ ~ ( graph d ) with parameters

(v, nl, d ) i s equivalent t o t h e existence of a symmetric (v, nl, d ) B I B design with t h e property Corollary.

(s).

The parameters (v, nl,

d ) of a regular ~ ~ ( graph d ) are

connected by t h e r e l a t i o n

3 - Necessary conditions f o r t h e existence of regular ~ ~ ( graphs. d ) Let A be t h e adjacency matrix of a regular g2(d) graph with parameters (v, nl, d ) .

Let J be t h e v

x

v matrix f o r which each element i s unity.

Then it i s r e a d i l y seen t h a t

Hence

= (nl

-d

) + ~nldJ

.

R. C. BOSE

*

Thus A has only t h r e e d i s t i n c t c h a r a c t e r i s t i c r o o t s n 1 and (nl

- d)'12.

Now from t h e r e g u l a r i t y of G2(d) it follows t h a t A* = A/nl

i s a s t o c h a s t i c matrix, and s i n c e G2(d) i s connected A* i s i r r e d u c i b l e .

It follows [ ~ r a u e r (1952)] t h a t u n i t y i s a simple r o o t o f A*, is a simple r o o t o f A.

el

L e t al, a 2 be t h e m u l t i p l i c i t i e s of t h e r o o t s

and O2 = -(nl - a ) 'I2.

= (n,

To determine

q

so t h a t nl

Then

and a we n o t e t h a t 2

Since t h e m u l t i p l i c i t i e s a r e n e c e s s a r i l y i n t e g r a l , t h e r e must e x i s t an i n t e g e r m such t h a t n

2

1

= d+m

.

Also s i n c e

2a2 = (V - I )

- (m+i)

d/m must be i n t e g r a l , and t h e i n t e g e r s v - 1 - m

and d/m must have t h e

same p a r i t y , I n p a r t i c u l a r l e t d = 1. Then m = 1 and from (7.2.,1), v = 3. Hence a r e g u l a r G ( a ) graph with d = l m u s t n e c e s s a r i l y be a complete graph of 2 o r d e r 3.

Taking t h i s together with Lemma (7.2.1) we have t h e theorem of

Erdtis , Renyi and S6s (1966 ) Theorem (7.3.1).

.

A f i n i t e graph G i n which any two d i s t i n c t v e r t i c e s

a r e simultaneously adjacent t o e x a c t l y one v e r t e x , c o n s i s t s o f n sub-

R. C. BOSE graphs of order 3, which have a common vertex when n a 2 . Thus G2(d) graphs with d = 0 or 1 are canpletely characterized.

For

d i 2 we have the following theorem: I f G2(d) i s a f i n i t e graph without loops or multi-

Theorem (7.3.2).

ple edges,in which each pair of d i s t i n c t vertices is adjacent t o exactly d other vertices, d 2 2 , then G2(d) i s regular of valence nl such t h a t v - 1 = n (n - l ) / d 1 1

where v is the number of vertices and there exists a

positive integer m, such t h a t 2 ( i ) nl = d+m ( i i ) d/m i s en integer, with the same parity a s v

- 1-m.

We w i l l next address ourselves t o the problem of construction of G, (a ) graphs.

4.

G2(d) graphs derived fran p a r t i a l geometries.

1 geometric graph ( r , k, t ) , with parameters v, nl, pll,

Consider a pseudo 2 pU given by

(4.4.7) and (4.4.4). When the condition k = r + t + 1 is s a t i s f ied we have 1 2 pll = pU = rt. Since the graph of a p a r t i a l geanetry is also pseudo geanetric any p a r t i a l gecmnetry f o r which k = r + t + 1 w i l l provide a G2(d) graph.

Many examples of p a r t i a l geometries satisfying these conditions

are known. ( a ) Consider en e l l i p t i c non-degenerate quadric i n the f i n i t e projective space PG(5, q) whgre q is a prime power.

It is kaavn [ ~ r k o s e

(1957) and Ray-Cheudhuri (1961 a)] t h a t t h i s quadric i s ruled by straight lines called generators, but contains no planes. the surface is (q3 + l ) ( q2 +I).

The number of points on

It was shown i n Bose (1963 c ) , t h a t the

points and generators can be regarded as the points and lines of a par2 tial geometry (q + 1 , q + l , 1). The dual of t h i s partial geanetry i s obtained by taking the points and l i n e s of the dual t o be the l i n e and

points of the original gecmetry. parameters ( q + 1, q2 + 1, 1).

Thus the dual p a r t i a l gecmetry has the

Another

w

t o obtain a p a r t i a l geaetrY

with the same parameters is t o take the -face

= 0 i n N ( 3 , q2).

X ~ + l + ~ ~ + l + ~ ~ + l + ~ ~ + l

It has k e n shown in Bose and Chakravarti (1966) that

t h i s surface contains (q2 + 1 )(q3 + 1 ) points, and (q + 1 )(q3 + 1 ) generators which may be taken t o be points and l i n e s of a partial geametry 1 ( q 1 q2 + 1 1 I f ve take q a 2, the condition pll = p L i s satisfied. Tbe graph of t h i s p a r t i a l g e a e t r y is strongly regular with parameters

(45, 12, 3, 3).

~ h u swe get a G ~ ( &with ) v = 45, nl = 12, d = 3.

Here

m = 3. (b) For a pseudo net graph ~ ~ ( with k ) parameters (6.5.1) t h e condition k = r + t + lreduces t o k = 2r. a G2(d) graph with v = 4r

Thus a pseudo net graph ~ ~ ( 2 is r )

,% = r ( & - l ) , d = r ( r - 1 ) .

Wecanthusget

~ ~ ( graphs d ) f o r all values of r for which there a i s t r - 2 mutually orthogonal Latin squares of order 2r. where m is a non-negative integer. with

= 22"'+2,

5=

This is elways true i f r

2m

W e can therefore obtain a G2(d) graph

- 11, a = P(2.- 1).

Again since t

a Latin square of order 6, we get a ~ ~ ( graph d ) with v = 36, d = 6.

a

h exists

5 = 15,

Also the existence of 5 mutually orthogonal squares of order 12

is known [Bose, Chakravarti, and .Krmth (1960)l.

By taking 4 mutually

orthogonal squares of order 12, we can get a ~ ~ ( graph d ) with v = 144,

4=

66,

a

= 30.

( c ) The dual of a design is defined as a new design whose treatments and blocks are i n (1, 1 ) correspondence with the blocks and treatments of the origindl design, and incidence is preserved (a block and treatment are incident i f t h e treatment i s contained i n the block and non-incident otherwise).

It i s bnawn [Bose (1963 c )] t h a t the dual of a BIB design

R.

C. BOSE

with parameters vO, bo, ro, kg, Xo = 1 can be regarded a s p a r t i a l geme t r y ( r , k, r ) where r = kg, and k = ro, t h e l i n e s of t h e p a r t i a l geome t r y being t h e blocks of t h e dual design. Such a dual design ( o r p a r t i a l geometry) may be c a l l e d a linked block design (or geometry).

The corresponding strongly regular graph

has t h e parameters (7.4.1)

v = k(kr-k+l)/r,

(7.4.2)

p1 = k - 2 + r 11

nl = r ( k - 1 )

,

2

.

,

2 pll

= r2

It w i l l be called a linked block graph and w i l l be denoted by m r ( k ) . Any strongly regular graph (not necessarily t h e graph of a linked block design) w i l l be c a l l e d a p s d o linked block graph LBr(k). k = r + t + l now reduces t o k = 2 r + l . %(2r+l)

ko = ?-I, A.

nl = 2r2, d = r

v0* * ? -'{a - I ) , bo = 2a

2

.

Ila BIB

- 1, ro = 2'+ 1,

= 1 are known [ h s e and Shrikhande (1960 b ) ] f o r every

i n t e g r a l value of m. with v = 2*

Thus a pseudo l i n k e d block graph

i s a G2(d) graph wftb v = 4 r 2 - I ,

designs with parameters

The conditiotl

- 1, nl

We can theref&

= 2p-',

ti = 2a-2

g e t a corresponding G2(d) .- graph f o r all i n t e g r a l m.

Also BIB

designs with parameters (7.4.3) vo = r ( 2 r - I ) ,

bo = b r 2 - 1 ,

r o =2 r + l ,

k O = r,

AO = 1

a r e lmown f o r values of ko = 2, 3, 4, 5 and 7 [ ~ o s e(1939) and Rao (1961)l Hence t h e corresponding G ( d ) graphs with parameters v = 4r 2

2

- 1, nl

= 2r2,

d = r2 can be constructed f o r r = 2, 3, 4, 5 and 7.

5.

.-

G2(d) graphs of negative Latin square type.

A strongly regular

graph corresponding t o Mesner's negative Latin square association scheme [Chapter 111, paragraph 3 ( e ) ] . w i l l be called a negative Latin square

R. C.

BOSE

graph I L ~ ( ~ ) .Its parameters a r e

I f k = 2 r , then

= p2 = r ( r - 1 ) . Hence a negative Latin square 11 11 graph NLr(2r) i s a G (d) graph with parameters v = 4r 2 * nl= r ( 2 r + l ) , d = r(r+l). We shall now show t h a t a negative Latin square graph NL ( 2 r ) can be

r

derived from a n e t graph L ( 2 r ) . net of degree r and order 2r.

A n e t graph Lr (21-1 i s t h e graph of a

Take any c l a s s C of p a r a l l e l l i n e s i n t h e

net, and divide them i n t o groups of r l i n e s each. v e r t i c e s corresponding t o t h e 2r

2

Let V1 be t h e s e t of

p o i n t s on l i n e s of t h e f i r s t group, and

V be t h e s e t of v e r t i c e s corresponding t o t h e 2rL points on t h e l i n e s of 2

t h e second group.

I f P i s a point on a l i n e

of t h e f i r s t group, then

t h e vertex x corresponding t o P i s adjacent t o t h e 2r-1 v e r t i c e s corresponding t o t h e other points on l i n e s o t h e r than than C).

!. (one belonging

- 112 i n t e r s e c t i o n s a r e 1

adjacent t o x. 2

w2 = r v e r t i c e s i n V2. 2nl

-

t o each of t h e p a r a l l e l c l a s s e s other

1 (v/2) = pU

We t h u s g e t wl = r 2

It i s c l e a r t h a t these a r e a l l t h e v e r t i c e s

Similarly each v e r t e x i n V2 i s adjacent t o exactly Again f o r t h e net graph ~ ~ ( 2 r ) ,

+ pll2 = 2 r ( r

(4.2.1) a r e s a t i s f i e d .

The v e r t i c e s corresponding t o t h e s e

adjacent t o t h e vertex x.

v e r t i c e s i n V adjacent t o x. 1 in V

Also through P t h e r e pass r - 1

Each of t h e s e l i n e s i n t e r s e c t s each l i n e of t h e f i r s t group

(other than a ) i n a s i n g l e point. (r

e.

- 1).

Thus t h e conditions of Theorem

Hence by complementation with respect t o V1 and V2

we obtain a s t r o n g l y regular graph with parameters

R. C . BOSE lr 2r pl1 = r ( r + I ) = pll

.

This i s a negative Latin square graph NLr(2r) by d e f i n i t i o n .

We

t h u s have A negative Latin square graph NLr ( 2 r ) e x i s t s when-

Theorem (7.5.1).

I n p a r t i c u l a r a negative Latin square

ever a n e t graph ~ ~ ( 2 erx i)s t s .

graph NLr(2r) e x i s t s f o r r = 3, 6 and

6.

zn-'

where n > 1.

Composition of pseudo net and pseudo negative Latin square graphs.

W e have seen t h a t when t h e incidence matrix B = (niJ ) of a symmetric BIB design (vo, kg, A ) s a t i s f i e s t h e conditions ( s ) , n = n for i # J , 0 ij ji n.. = 0 then B i s t h e adjacency matrix of a G ~ ( xgraph. ~ ) We w i l l c a l l 11

N, an incidence matrix of type I.

Let J denote t h e v0x v

0

matrix all of whose elements-are unity,

N i s t h e incidence matrix of a symmetric BIB design (vo, ko

-N = '5- N then (vo, v0

Ao)

If

and

i s t h e incidence matrix of a symmetric BIB design

- koy vO- 2k0 + AO).

I f N i s of type I, then

each element of t h e main diagonal i s unity.

f is

symmetric, and

The incidence matrix of a

symmetric BIB w i l l be defined t o be of type I1 when it s a t i s f i e s these conditions. Consider a c l a s s Q of symmetric BIBdesigns (vo, kg, Ao) f o r which t h e condition vo/4 = ko

- Xo

is satisfied.

Then t h e following r e s u l t

which we s t a t e i n the form of a lenrma i s known [Shrikhande (1962)l. Lemma (7.6.1).

I f Nl and N;

symmetric BIB designs (vl, kl,

a1 )

a r e the incidence matrices of two and (v2¶ k2, ),2) belonging t o t h e c l a s s

Q , then

- -

N = NlxN 2 +NlxN2

i s t h e incidence matrix of a symmetric B I B design (vo, kg, XO) belonging

R. C . BOSE

to

a

where

and

denotes t h e Kronecker product. We a l s o note t h a t i f N

1

i s of type I1 and N

2

i s of type I, then N i s

of type I. I f t h e adjacency matrix of a ~ ~ ( graph 6 ) with parameters, v., nl, d i s t h e incidence matrix of a BIB design of t h e class S i n c e v - 1 = n (n - l ) / d , 1 1

then v = r ( n l - d ) .

it follows t h a t nl = $ [ ( 4 d + l ) f ( h d + l ) 1/21,

which shows t h a t nl = r(2r + 1 ) o r nl = r ( 2 r

r.

a,

- 1 ) f o r s F e p o s i t i v e integer

Thus G2(d) must be e i t h e r a pseudo net graph Lr(2r) o r a negative

Latin square graph L ( a ) .

r

Theorem (7.6.1).

The existence of pseudo net graphs %,(2r

) and 1

~ ~ ~ (implies 2 r ~ t h)e existence of a pseudo net graph ~ ~ ( 2 1with -1

Let N1 be t h e adjacency matrix of t h e pseudo net graph Lr (2rl) and 1

N2 t h e adjacency matrix of t h e pseudo net graph Lr !2r2).

2 incidence matrix of a symmetric BIB [4r1,

Then N1 i s t h e

- 11, rl(rl - 111 and -N2 13 c

rl(2rl

2 i s t h e adjacency matrix of a symmetric BIB [4r2, r 2 ( 2 r 2 + 1 ) , r 2 ( r 2 + l ) ] where N

1

i s t o type I and

r2i s of type 11. From Lemma (7.6.1) N = N 2 x N +N2xN1

,

1

i s t h e adjacency matrix of a symmetric BIB (vo, kg, Ao), belonging t o

a,

where

Since N i s of type I, it iollows t h a t it i s the adjacency matrix of

K. C . BOSE

a pseudo net graph L ( 2 r ) , where r = 2 r r r 1 2' Theorem (7.6.2).

A pseudo n e t graph Lr(2r) e x i s t s f o r all

r = 3 m - P-l, where m and n a r e non-negative i n t e g e r s , (my n ) # ( 0 , 0 ) . A n e t graph i s a l s o a pseudo net graph.

Lr(2r) e x i s t s f o r r =

zn-',

n $1. Hence t h e theorem i s t r u e f o r m = 0 ,

Again a net graph ~ ~ ( 2 erx i)s t s f o r r

ns1.

a pseudo n e t graph L~( 2 r ) f o r r =

3m-l

.?",

graph 4 ( 2 r ) e x i s t s f o r r =

6

f o r r = f-I

t e n c e of a pseudo net graph

by c h o o s i a rl =

Hence a pseudo n e t graph

9.2m-1

r2 = 3.

-

f 9-'.

0 3. 2

.

?-2,

Assuming t h e exist h e existence of

follows from Theorem (7.6.1)

Hence by induction a pseudo net

Thus t h e theorem holds f o r

ma 1,

n = 0. F i n a l l y t h e existence of t h e pseudo n e t graph ~ ~ ( 2 f ro r)

= 3m. 2m+n-1 follows from Theorem (7.6.1) by choosing rl = 3m and r = 2n-1, 2

where m > 1, n > 1. Hence t h e theorem holds f o r m & 1, n

Corollary, d = r(r

zm-l

G ( d ) graphs with parameters r = 4r2, nl = r ( 2 r 2

- 1 ) e x i s t f o r a l l r = 3m

i n t e g e r s and (my n )

- 1),

2m+n-1 where m and n a r e non-negative

# (0, 0 ) .

Similarly we can prove [ ~ o s eand Shrikhande (1970)l t h e -fGllowing theorems : The existence of a pseudo n e t graph L (2rl) and rl a negative Latin square graph % (2r2) implies t h e existence of a neg2 a t i v e Latin square graph NLr ( 2 r ) , where r = 2rlr2. Theorem (7.6.3).

Theorem (7.6.4).

A negative Latin square graph ~ ~ ~ ( e x 2 i srt s) f o r

a l l r = 3m ?+"-l,where m and n a r e non-negative i n t e g e r s , (my n )

# (0, 0 ) .

Corollary.

~ ~ ( graphs d ) with parameters v = 4r

2

, nl

= r (2r + 1 ) ,

d = r ( r + 1 ) e x i s t f o r a l l r = 3m. P-lwhere m and n a r e non-negative

1

R. C . BOSE

integers and (my n) # (0, 0).

7. Descendant of a stromly regular graph. Let G be a strongly regular graph with parameters

and vertex set XO, X1, where Vo = {xl,

..., x

. a * ,

Xn19

Xnl+ls

and V2 = {xn 1 vertices adjacent and non-adjacent to xo. }

9

"

'*

XV-l '

..., xv-1) are the sets of If Go is the subgraph of G

which remains by deleting xo and the edges incident with it, and G, is obtained fromGOby complementation with respect to V1 and V2 then G, is defined to be

the descendent of G with respect to the vertex x. Bose

and Shrikhande (1971)have proved

V,

Theorem (7.7.1). If G is a strongly regular graph with parameters 1 2 nl, pll, pll, the necessary and sufficient condition for the descend-

ant G, of G (with respect to any vertex xo) to be strongly regular is

When this condition is satisfied the parameters of G, are given by (7.7.2)

2 v, = v-1, nl, = 2sl-2pll

,

8. G2(d) graphs derivable as descendants. Consider a strongly 1 2 regular graph G with parameters v, nl, pll, pll, for which the condition

(7.7.1) is satisfied. Then its descendant G, has the parameters (7.7.2) and (7.7.3).

1 If G, is a ~~(d,)graph, pll,

--

2 pll,.

Thus G itself must be a a2(d) graph, for whichl:p

1 Hence pll = = l:p

= d.

2

Substitut-

R. C. BOSE ing i n (7.7.1) we have v = 4(nl-d).

It follows t h a t G must either be a

pseudo net L~ (2r) graph or a negative Latin square

( 2 r ) graph.

We are

therefore lead to studying descendants of such graphs. Theorem (7.8.123

The descendant of a pseduo net graph ~ ~ ( 2or r )a

negative Latin square graph ~ ~ ( 2is r )a pseudo linked block graph L ~ ~ (+21r) . A pseudo linked block graph q ( 2 r + 1 ) exists f o r all r /

=

3. p-'

where m and n are non-negative integers (m, n) # (0, 0). L e t t h e parameters of a pseudo net graph ~ ~ ( 2be r )v = 4r 2 ,

n = r(2r 1

- l ) , pU1

-

2 = pU = r ( r 1 ) . The condition (7.7.1) i s satisfied

and the parameters of the descendant graph G, given by (7.7.2) are v* = 4r2 -1,

(7.8.2)

1 2 p * = 3 .P,*=P,*=r

2

Thus G, is by definition a pseudo linked block graph LBr(2r + 1). It can be proved exactly in the same way that the descendant of a negative Latin square graph I,=(&), Corollary.

has pracisely the partmeters (7.8.2).

-

G2(d) graphs with parameters v = 4r2 1, nl =

exist f o r all r = 3m P-lwhere m and n are non-ative

d, d = r 2

integers

(m, n) # (0, 0 ) .

9.

Ascendent of a 8trongly regular graph.

1 2 regular graph with parameters (v, nl, pU, pU).

L e t G be a strongly Let (vl, v2) be a

partition of the v e r t u s e t V of G where V1 and V2 respectively contain n.1 and V-n; vertices. graph with vertu s e t The v e r t u of vl).

-

Let (-,

-

V).

be a vertex not i n V and l e t G* be a

We define adjacency i n G* a s follows:

is adjacent only t o vertices of V1 ( and t o all vertices

I f x, y are i n V, then they are adjacent i n G* i f and only i f

they are adjacent i n G and belong both t o V1 or both t o V2, or i f they

are nonadjacent i n G and belang one t o V1 and the other t o V2.

If t h e

l* pu) 20 where v' graph G* i s strongly regular' with parameters (v', ny , pus

i s necessarily v + l , then G* i s said t o be an ascendant of G. Bose and Shrikhande (1971) have derived the conditions unde-ch 1 a graph G with parameters (v, nl, pu, l* 2* parameters (v*, n t , pU, pU)

.

2 pll)

has an ascendent G* with

They prove

Theorem (7.9.1). Let G be a strongly regular graph wi%hparameters 1 2 (v, nl, pU, pU). Then G has an ascendent G* with parameters

(I.,nfi, p g , p g ) i f and only it the following parametric and structural conditions (P) and (S) are satisfied i n G. 2 1 (P) v = 6pU-2pll-1. (S)

The equation

has an integral solution nff and there e x i s t s a partition (V1, V2) of the 1

vertex s e t V of G with nfi vertices i n V1 and v-n:

vertices i n V2 such

that every vertex i n V1 has nfi- nl+pU1 adjacent vertices i n V1 and 2 every vertex i n V has pU adjacent vertices i n V2. 2 The parameters of G* are then given by

It i s obvious t h a t any two blocks of a BIBD with X = 1 have a t most

one treatment i n common.

Consider a BIBS with r = 2 k + l , A = 1. Then 2 the values of v and b are given by v = 2k2 k and b = 4k 1. Consider

-

-

the blocks a s vertices of a graph G and define two blocks a s adjacent or nonadjacent according as they have a treatment in ccnrmon or not. Then 2 2 [2] G i s strongly regular with parameters (4k 1, 2k2, k2, k ) and sat-

-

i s f i e s the condition (P) of the above theorem.

Also the equation

R. C. BOSE

-

f (x) = 0 has integral solutions k(2k - 1 ) and k(2k + 1 ) . Take n: = k(2k 1).

-

I f the 4k2 1 blocks can be partitioned into s e t s V1 and V2 of k(2k - 1 )

-

and (k + 1)(2k 1 ) blocks respectively such that each block i n V1 i s ad-

-

jacent t o k2 k blocks i n V1 and each block i n V2 i s adJacent t o k2 blocks i n V2, then the condition ( s ) i s also satisfied.

From t h i s

it follows t h a t the s e t V1 (respectively v2) contains each of the 2k2 - k

treatments exactly k (respectively k + 1 ) times. We note t h a t a BIBD with r = 2 k + l , A = 1 i s a p a r t i a l gemetry ( r , k, t ) = (2k+l, k, k).

The graph G i s then the graph of the dual

configuration and i s also a p a r t i a l geometry (k, 2 k + l , k).

We can,

therefore, s t a t e the following theorem. Theorem (7.9.2).

L e t G be the graph of the dual of a BIBD with

r = 2 k + l , A = 1. Then G has an ascendant G* which i s a pseudo ~ ~ ( 2 k )

graph i f and only i f the 4k2 -1 blocks of the BIBD can be partitioned

-

-

into s e t s V1 and V2 of k(2k 1 ) and (k + 1 )(2k 1 ) blocks respectively

-

such t h a t each of the 2k2 k treatments of the BIBD occur k times i n V1 and k + 1 times i n V2. BIB designs having the structure of the above theorem exist for

k = 5 and 7 C ~ a l (1967), l Appendix

11.

Hence we have the following re-

sult Corollary.

Pseudo L (10) and pseudo % ( l 4 ) graphs exist. 5 Goethals and Seidel (1970) have constructed a pseudo ~ ~ ( 1 graph 0) i n precisely the same manner. Using Theorems (7.6.6.1,

(7.6.21,

(7.6.3) and t h e i r corollaries we

have the following theorems Pseudo Lr(2r) graphs exist f o r all Theorem (7.9.3). m a c m+a+c+n-1 r 2 3 5 7 2 where m, n, a, c are non-negative integers

-'Theorem'(7.9.4).

N L ( 2 r ) g r a p h s e x i s t f o r r = 5 a c7 2a+c w h- e r e , a , c

r

a r e non-negative i n t e g e r s and f o r r = 3m587c~+a+c+n-1 where m, n, a , c a r e non-negative and (my n )

# (0, 0).

Finally, noting t h a t pseduo L ~ ( P I - )and ~ ~ ~ (graphs 2 r ) a r e G2(d) graphs we have Theorem (7.9.5). (i) (ii)

v = 4r

*

G2(d) graphs with t h e following parameters e x i s t

= r ( 2 r - 1 1 , d = r ( r -1);

2

v = 4 r -1, nl = 2r2, d = r2;

for a l l r =

3m59c2m+a+c+n-1 where m, n, a , c a r e

non-negative i n t e g e r s

( i i i ) v = 4r2, nl = r ( 2 r + I ) , d = r ( r + I ) ;

c a+c , where a , c a r e non-negative integers and with with r = 5 9 2 ,

= 3m5y,c~+a+c+n-1 where m, n, a , c a r e non-negative i n t e g e r s and

We remark t h a t since our construction i s e s s e n t i a l l y b y a composit i o n method, any new ~ ~ ( graph d ) with parameters a s i n c o r o l l a r i e s t o Theorems (7.6.2), t h e above theorem

(7.6.4) o r (7.8.1) can be u t i l i z e d i n conjunction with t o enlarge such a family considerably.

CfFAPTER VIII

GEOMCrmIC AND PSEUDO GEOMETRIC GRAPHS (q2 + 1, q + 1, 1 ) 1. The desinn corresponding t o a vertex of a strongly regular graph.

Let A be t h e adjacency matrix of a strongly regular graph with parameters

R. C . BOSE 1

v, nl, pll,

2 pll.

Noting t h a t A is the association matrix B1 of the cor-

responding association schene, we have from (3.4.5)

where Iv i s unit matrix of order v and J i s a v x v matrix with each V

element unity. Let OO.be a particular vertex of G.

Let G be the nlxn2 suhnatrix

..., 8nlwhich are adand whose ~olumnscorrespond t o the vertices B1, B2, ...,

of A, whose rows correspond t o the vertices 01,02, $went t o 00,

which a r e non-adjacent t o OO. row of B corresponding t o Oi,

1

since there are p12 vertices among B1,

..., Bn2 which are adjacent t o both

B which corresponds t o B

vertices among 8

e2,

5

I

There are exactly p12 unities i n the

and

el.

B29

Similarly the column of

2 2 has exactly pll unities, since there are pll

..., 8nlwhich are adjacent t o both

O0 and B

5'

Hence B i s the incidence matrix of a design with psrameters v' = nl, 1 2 b' = n2, r' = p12, k' = pU. We shall say t h a t t h i s design corresponds t o the vertex 00, and denote it by D(eo). 2.

The ~ r m r t i e s(P) and (PL.Let G be a pseudo geometric graph

( r , k, 1). Then G w i l l be said t o have the property (P)with respect t o the vertex 00, i f the r ( k - 1 ) vertices adjacent t o O 0 can be partitioned into r disjoint s e t s S1, S2,

..., Sr of

size k - 1 such that any two ver-

t i c e s belonging t o the same s e t Si are adjacent. clique.

Then Ki = SiUOO i s a

Since t = 1, any vertex 8 whichbelongsto Si i s non-adjacent iu

t o any vertex O'ilul

which belongs t o Sit, i # i t .

We shall say that G has the additional property (P") i f any t m vert i c e s Bj,

B j l both non-adjacent t o go, and both adjacent t o OiU and 0 i 'u'

are thenselves non--adjacent, where OiU

and 8i,U, a r e sny two vertices be-

R. C. BOSE longing t o Si and Si, respectively, i f i f . 3.

The design D(Oo) corresponding t o a vertex go of a pseudo geo-

metric graph (q2 + 1. q + 1, 1 ) .

Let D(e0) be the design corresponding t o

2 t h e vertex go i f a pseudo-geometric (q + 1, q + 1, 1 ) graph G, wbich,has The parameters of G a r e

the property (P) with respect t o t h e vertex go.

Be definition t h e treatments of l)(eO) correspond t o t h e vertices of G adjacent t o 00, and t h e blocks of O(eO) correspond t o t h e v e r t i c e s of G non-adjacent t o go.

Hence t h e number of treatments i s

2 4 v' = n1 = q(q + I ) , and the number of blocks i s b' = n2 = q

.

Using t h e property (P) Bose and Shrikhande (19'72) have shown t h a t l)(eO) i s group d i v i s i b l e (GD) design with parameters

Since X;vf-r'k'

= 0, t h e design i s semi-regular.

contains exactly one treatment from each s e t . Theorem (8.3.1).

Hence each block

W e therefore have

2 I f G i s apseudo-geametric graph ( q +1,q+l, 1 )

having the property (P) with respect t o t h e vertex

eO,

then t h e design

D(OO) corresponding t o the vertex go i s a semi-regular group d i v i s i b l e

(SRGD) design, with parameters (8.3.3). Corollary.

Each vertex non-adjacent t o

one vertex i n each of t h e s e t s S1,

s2,

.

- S

eO,

Sq2+y

i s adjacent t o exactly hro blocks of ~ ( 9 ~ )

w i l l be called f i r s t associates i f they correspond t o two v e r t i c e s i n B which a r e adjacent, and second associates i f they correspond t o twu ver-

R. C. BOSE

t i c e s i n B which a r e non-adjacent.

Since each vertex i n B i s adjacent t o 2 2 p12 other v e r t i c e s i n B, and non-adjacent t o p22 other v e r t i c e s i n B, each block of D ( B ~is ) f i r s t associate of

m i blocks

and second associate of

m; blocks where

Under t h e hypothesis of Theorem (8.3.1) we can now further prove [Flose and Shrikhande (197211 t h e following Corollaq.

I f a treatment occurs i n a block 0

3

of D(o0), then it

occurs q - 1 times among t h e blocks which a r e f i r s t associates of 0 qJ

- q times among t h e blocks which a r e second associates of

0

3'

j'

and

If a 2

treatment does not occur i n a block 0 of D(OO) then it occurs q times 3 2 among t h e blocks which a r e f i r s t associates of 6 and q3 - q times among

3

t h e blocks which a r e second associates of 0

3'

Let us now assume that G has t h e additional property (P*). We s h a l l show t h a t under t h i s hypothesis any two blocks of D(O0) i n t e r s e c t i n one treatment i f they a r e first associates and i n q + l treatments i f they a r e second associates. 8i,ul.

Let t h e block 0

contain two treatments 8 -and j Iu They must belong t o d i f f e r e n t s e t s . Hence t h e vertex BiU belongs

t o Si and t h e vertex O i q u I OiIu,

occurs i n a block 0

belongs t o Sit

d'

perty (P*) i s contradicted.

,i #

i'

.

I f the pair

which i s a f i r s t associate of 0

d'

eiu,

then pro-

Hence two blocks of ~ ( 8 which ~ ) are f i r s t

associates cannot have more than one treatment i n common.

Since each

the occurs q - 1 times among t h e f i r s t associates of 0 j 5' number of blocks which are.$irst associates of 0 and contain exactly one treatment of 6

treatment of 0

3

-

i s (q2 + l ) ( q 1 ) .

a s s o c i a t e s of 0 j'

3

But t h i s i s t h e t o t a l number of f i r s t

Hence any two blocks of ~ ( 8 which ~ ) a r e f i r s t associ-

R. C . BOSE ates intersect i n exactly one treatment. 2 Let us now consider the dibtribution of the k' = q + 1 treatments belonging t o t h e block 6 of D(eo) among t h e j

associates of 8

where

.la

mi

m i blocks which are second Let xi be the number of

i s given by (8.3.4).

treatments i n which 6 intersects the i-th block which i s a second asso-

.l

c i a t e of BA, i = 1, 2,

..., m'.

By the second corollary t o Theorem

-

(8.3.1) each treatment of 6 occurs q3 q among the .second associates of

Bj.

Again any pair of treatments belonging t o 6 must occur A;

3

times among the second associates of 6

3'

f i r s t associates.

- 1 = qZ - 1

since it cannot occur among the

Hence

i=l

Let

Hence

This shows t h a t xi

- x- = 0,

i.e.

, any

two blocks of V(e0) which are

second associates, intersect i n exactly q + l treatments. Theorem (8.3.2).

2

I f the pseudo geometric graph ( q + 1, q + 1, 1) of

Theorem (8.3.1) has the additional property (P) with respect t o the vertex

eO,

then the design D(eo) corresponding t o the vertex O 0 has the

property (I1) t h a t any two blocks which a r e f i r s t associates intersect in exactly one treatment and any two blocks which a r e second associates

intersect i n q + l treatments. We can now further p r w e [ ~ o s eand Shrikhande (197211 the following 2 I f G i s a pseudo geanetric graph (q + 1, q + 1, 1 ) Theorem [8.3..3). having the properties (P) and (PI) with respect t o the vertex

eO,

then

the subgraph G2 of G, whose vertex set i s the s e t of those vertices of G which a r e non-adjacent t o 00, i s a strongly regular graph with psram-

eters

4

(8.3.5)

.

i e.

v* = q ,

-

n

2

+

P

-

Pz=q(q-l)

9

,a

negative Latin square graph NL ( q2 1. q-1 Corollary. The dual P ( e 0 ) of D(eO) i s a p a r t i a l l y Wanted inccm-

plete block (PBIB) design based on a negative Latin square association scheme NL (q2). q-1 4. Semireuular n o u n divisible designs (SRGD)with the DroDerty (I1). -

The converse of Theorem (8.3.3) i s of great interest.

Given an

SRGD design 0 with psrameters (8.3.31, and having the additional property (I1) t h a t any two blocks of 0 intersect i n either 1 or q + l treatments, ye can ask whether there e x i s t s a pseudo geometric graph G having t h e

properties (P) and (P*), with respect t o a vertex O0 of G such t h a t D i s isomorphic t o D(OO). The answer is i n the affirmative as has been proved by Bose and Shrikhande (1972), but as the proof i s quite long we s h a l l merely s t a t e the theorem obtained by them. Theorem (8.4.1).

Given an SRGD design D with parameters (8.3.31,

having the property (I1) t h a t any two blocks of 0. intersect i n either 1 2 o r q + 1 treatments, there exists a pseudo geometric (q + 1, q + 1, 1 ) graph G, and a vertex

e0

of G such that D i s i s a o r p h i c with the design

o(eo) corresponding t o eO, and G has the properties (P) and (PI) with

R. C . BOSE

respect t o 8

0' I f a treatment occurs i n any block 0 of

Corollaq. occurs q

- 1 times among t h e f i r s t associates of

I'

8

j

and q3

D,

then it

- q times among

t h e second a s s o c i a t e s of 0 If a treatment does not occur i n 0 then J' 3' 2 2 it occurs q times among the first associates of B j and q3 - q times among t h e second associates of 13 3' 5. P a r t i a l geometries and geometric ~ r a p h s(q2 + 1, q + I., 1). Let

P be

2

a p a r t i a l geometry (q

+ 1, q + 1, 1 ) .

The points and l i n e of

P

sat-

i s f y axioms A 1 - ~ of 4 ChapteF IVY paragraph 3 with r = q2 + 1, k = q + 1,

t = 1. Let G be t h e graph of t h e geometry, then G i s strongly regular with parameters given by (8.3.1) and (8.3.2).

Two points of t h e geometry

which a r e incident with t h e seme l i n e and a r e therefore adjacent i n G, may be c a l l e d adjacent points. Similarly tvo points of t h e geometry may be said t o be non-adjacent i f t h e r e i s no l i n e of the geometry incident with both.

They a r e non-adjacent i n G.

Let go be a vertex of G, i.e.,

a . (i=

1, 2,

...

a point of

, q2 + 1 ) be t h e l i n e s of

P

P.

Let

incident with go.

Let S dei

note t h e s e t of q points (other than g o ) incident with 1 % . Then S t h e s e t of points adjacent t o

sly s2' eO. i

#

...

Y

sq2+1

eO,

i s t h e union of the d i s j o i n t s e t s

and G obviously has t h e property (P) with respect t o

Any two v e r t i c e s belonging t o t h e same s e t S. a r e adjacent, but i f i t , then a v e r t e x belonging t o Si i s non-adjacent t o a vertex be-

longing t o Si,

.

It follows from t = 1 t h a t t h e r e cannot e x i s t a t r i a n g l e i n t h e

geometry, i . e . ,

i f Bl,

e2,

83 a r e any t h r e e d i s t i n c t points which a r e

pairwise adjacent, then they must be incident with t h e same l i n e a . From t h i s it e a s i l y follows t h a t G has t h e additional property (P*)with

respect t o

eO.

It now follows from Theorems (8.3.1) and (8.3.2) t h a t t h e

design D(8 ) corresponding t o t h e vertex 0 e t e r s (8.3.3),

e0

i s an SRGD design with param-

and possesses t h e property (I1) t h a t any tm blocks of Also frcm t h e corollary

D(o0) i n t e r s e c t i n e i t h e r 1 o r q + l treatments.

i s any treatment i n a block 6 of D(eo), then

t o Theorem (8.4.1), i f OiU

5

t h e r e a r e exactly q - 1 other blocks which a r e f i r s t associates of B

5

and contain t h e treatmen Biu.

We can then show t h a t i n t h e present case

D(8 ) has t h e additional property ( I 2 ) , t h a t t h e s e q 0

- 1blocks a r e mutu-

Hence we have t h e following theorem:

a l l y f i r s t associates.

I f G i s t h e graph of a p a r t i a l geometry

Theorem (8.5.1).

(q2 + 1, q + 1, I ) , then t h e design I)(BO)corresponding t o any vertex SRGD with parameters (8.3.3),

Consider any t h r e e point

e0

is

and possesses t h e p r o p e r t i e s (I1) and ( I 2 ) .

eO,

of t h e p a r t i a l gecuuetry

01,

Then 8 and 8 can be 1 2

(q2 + 1, q + 1, 1 ) which a r e pairwise non-adjacent.

i d e n t i f i e d with block of 0 ( e 0 ) , which a r e second a s s o c i a t e s and therefore i n t e r s e c t i n q + l treatments $ ,,, $1,

--ms

Since t h e treatments i n

OQ.

any block belong t o d i f f e r e n t s e t s , $i and $

3

a r e non-addacent-in G.

We

t h e r e f o r e have Corollary=

Given any t h r e e points

eO, el, e2

of a p a r t i a l geometry

(q2 + 1, q + 1, 1 ) which a r e pairwise non-adjacent, we can f i n d a s e t of q + 1 p o i n t s $0,

..., $Q which a r e pariwise non-adjacent eO,

which i s adjacent t o

8 and 1

e2.

We now consider t h e converse of Theorem (8.5.1). e x i s t s an SRGD design t i e s (II) and ( I 2 ) .

v

and each of

with-parameters (8.3.3)

Suppose t h e r e

and possessing t h e proper-

Then from Theorem (8.4.1) we can f i r s t construct a

strongly regular graph G with paremeters given by (8.3.1)

and (8.3.21,

and possessing a vertex B 0 with respect t o which G has t h e properties (P)

R. C . BOSE Theorem (8.5.2).

Given an SRGD design with parameters (8.3.3) and

possessing t h e properties ( I ~ )and ( I ~ ) , ,t h e r e e x i s t s a p a r t i a l geometry (q2 + 1, q + 1, 1 ) . such t h a t

D = D ( B ~where )

D(B0) i s t h e design corre-

s p o n d i ~t o some vertex of t h e graph G of t h e p a r t i a l geometry. 2 Now t h e p a r t i a l geometry (q + 1, q + 1, 1 ) i s known,

We have pointed

out i n Chapter V I I , paragraph 5 ( a ) two ways of obtaining it. Higman has shown (unpublished) t h a t t h e two r e a l i z a t i o n s a r e isomorphic.

It follows

from t h e theoran above Corollary.

I f q i s a prime o r a prime power, an SRGD design

0 with

parameters (8.3.3) and possessing t h e properties ( I ~ )and ( I ~ )e x i s t s . (q2) association scheme or graph e x i s t s . Also an I?L q-1 6. Unsolved problems and conjectures. Let eO, el,

e2

be any t h r e e

2 pairwise non-adjacent points of a p a r t i a l geometry ( q + 1, q + 1, 1 ) . Then we have shown i n corollary t o Theorem (8.5.1),

-..,$q2+1 which a r e pairwise non-erdjacent

s e t of q + l points $ ,,

and each of which i s adjacent t o 80, adjacent points $0, go,

el, e2,

t h a t there exists a

el,

€I2.

Now s t a r t i n g with t h e non-

$2 we may likewise obtain a s e t q + l points

.- - 0q2+1 , which a r e pairwise non-adjacent

i s adjacent t o $0, f o r all i, j = 1, 2,

$2.

and each of which

We may,conjecture t h a t 9 i s adjacent t o $ i 3

..., q 2 + 1 .

2 Again one may ask whether a p a r t i a l geometry ( q + 1, q + 1, 1 ) e x i s t s when q i s not a prime parer.

Even i f such a geometry does not e x i s t f o r

2 a given q, a pseudo geometric graph (q + 1, q + 1, 1)might e x i s t . I f q i s a prime power one may ask whether a l l p a r t i a l geometries (q2 + 1, q + 1, 1)a r e isomorphic.

I f t h e answer t o t h i s question i s i n

t h e affirmative, then our conjecture s t a t e d above i s c e r t a i n l y t r u e since it i s e a s i l y proved by geometrical considerations f o r t h e p a r t i a l

R. C . BOSE

geometry d e r i v e d from t h e e l l i p t i c quadric Q i n PG(5, q ) .

But t h e con-

j e c t u r e could s t i l l be t r u e even i f t h e r e e x i s t non-isomorphic p a r t i a l 2 geometries ( q + 1, q + 1, 1). Again l e t Ro a,nd R 2

e t r y (q + 1 , q + l , 1 ) .

1

be two non-intersecting l i n e s of a p a r t i a l g e m -

Let 900, go1,

gO2,

Through B O j t h e r e p a s s e s a unique l i n e m may denote by 6 the points

elO,

€JO0,

(j = 0, 1, 2,

elO,

i3 811,

...,

q0'

..., q).

J

..., 8oq be t h e p o i n t s of to.

meeting Ll i n

&

p o i n t which we

The condition t = 1 shows t h a t x

..., 9l q a r e all d i s t i n c t .

Let t h e p o i n t s of m

0 be Through any p o i n t of gi0 o f m0 t h e r e p a s s e s a unique

l i n e 1. meeting ml i n a p o i n t which we may denote by Oil,

( i = 0, '1, 2 ,

tinct.

e

i3

..., q).

The p o i n t s go1,

We may c o n j e c t u r e t h a t 1 . and m

( i , j = 0, 1,

..., q ) .

j

ell,

..., 9q l a r e a l l d i s -

intersect i n a point

Again t h i s c o n j e c t u r e i s c e r t a i n l y t r u e i f a p a r t i a l geometry (q2 + 1, q + 1, 1 ) i s always isomorphic t o t h e p a r t i a l geometry derived from t h e e l l i p t i c quadric Q ( a s can be proved by geometrical considera t i o n s ) but it could s t i l l be t r u e even i f non-isomorphic p a r t i a l geom2 e t r i e s (q + 1, q + 1, 1 ) e x i s t .

R. C. BOSE

BIBLIOGRAPHY Aigner , M. (1969). A characterization problem in graph theory. J. Comb. Theory 6, 45 - 55. Archbold, J. W. and Johnson, N. L. (1956). A method of constructing partially balanced incomplete block designs. Ann.Math. Statist. 27, 633 - 641. Bhattacharya, K. N. (1944). A new balanced incomplete block design. Science and Culture 9, 508. Bose, R. C. (1939). On the construction of balanced incomplete block designs. Ann.Eugenics 9, 353 - 399. (1942 a). A note on the resolvability of balanced incomplete block designs. Sankhya 6, 105 - 110. (1942 b) On some new series of balanced incomplete

.

block designs. Bull. Cal. Math. Soc. 34, 17 - 31. (1947 a). Mathematical theory of-the symmetrical factorial design. Sankhya 8, 107 - 166. (1947 b). On a resolvable series of balanced incomplete block designs. Sankhya 8, 249 256. (1949). A note on Fisher's inequaulity for balanced incomplete block designs. Ann. Math. Statist. 20, 619 620. (1952). A note on Nair's condition for pertSally

-

-

balanced incomplete block designs with k > r . Calcutta Statist. Assoc. Bull. 4, 123 126.

-

(1963 a). On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements. Calcutta Math. Soc. Golden Jubilee Commem. vol. (1958/59), part 11, 341 - 354. (1963 b). Combinatorial properties of partially balanced designs and association schemes. Sankhya 25, 109 - 136. Strongly regular graphs, partial geometries (1963 c ) . and partially balanced designs. Pacific J. Math. 13, 389 - 419. Bose, R. C. and Chakravarti, I. M. (1966). Hermitian varieties in a finite projective space PC (N, q2). Canad. J. Math. 18, 1161 1182.

R. C. BOSE Bose, R. C., Chakravarti, I. M. and Knuth, D. K. (1960). On methods of constructing sets of. mutually orthogonal Latin squares using a computer. I. Technometries 2, 507

- 516.

Bose, R. C. and Clatworthy,W..H. (1955). Some classes of partially balanced designs. Ann. Math. Statist. 26, 212 232.

-

Bose, R. C., Clatworthy, W. H. and Shrikhande, S. S. (1954). Tables of partially balanced designs with two associate classes. N. C. Ag. Expt. Station Tech. Bull. 107, Raleigh, N. C. Bose, R. C. and Connor, W. S. (1952).

Combinatorial properties of

group divisible incomplete block designs. Ann. Math. Statist. 23, 367 - 383. Bose, R. C. and Dowling, R. A. (1971). A generalization of Moore graphs of diameter two. J. Comb. Theory 11, 213 - 226. Bose, R. C. and Lasker, R. (1967). A characterization of tetrahed-

-

ral graphs. J. Comb. Theory 3 ,366 385. Bose, R. C. and Mesner, D. M. (1959). On linear associative algebras corresponding to the association schemes of partially balanced designs. Ann. Math. Statist. 30, 21 - 38. Bose, R. C. and Nair, K. R. (1939). Partially balanced incomplete block designs. Sankhya 4, 337 - 372. Bose, R. C. and Shimamoto, T. (1952). Classification and analysis of partially balanced incomplete block designs with two associate classes. J. Amer. Statist. Assoc. 47, 151 Bose, R. C. and Shrikhande, S. S. (1960 a).

- 184.

On the composition of balanced incomplete block designs. Canad. J. Math. 12, 177 - 188. (1960 b). On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler. Trans. Amer. Math. Soc. 95, 190 - 209. (1970). Graphs in which each pair of vertices is adjacent to the same number d of other vertices. Studia Sci. Math. Hungarica 5, 181 - 195. (1971). Some further constructions for G~(d) graphs. Studia Sci. Math. Hungarica 6, 127 - 132. (1972). Geometric and pseudo geometric graphs (q2+1, q + l , 1). J. Geometry 211, 75 - 94.

R. C. BOSE

Embedding t h e complement of an oval i n a pro-

(1973).

j e c t i v e plane of even order. Bose, R . C . ,

J . of Discrete Math.

Shrikhande, S. S. and Bhattacharya, K. N.

Math. S t a t i s t . 24, 167 Brauer, A. (1952).

Ann.

matrix. 75

- 195.

Limits

On

(1953).

t h e construction of group d i v i s i b l e incomplete block designs. f o r t h e c h a r a c t e r i s t i c roots of a

I V : Applications t o stochastic matrices.

Duke Math. J . 1 9

- 91.

Bruck, R. H. (1963).

F i n i t e nets 11. Uniqueness and imbedding.

P a c i f i c J . Math. 13, 421

- 457.

Bruck, R. H. and Ryser, H. J. (1949). f i n i t e projective planes. Chang Li-Chien (1959).

-

- 93.

The uniqueness and non-uniqueness of t r i -

angalar association schemes. 604

The nonexistence of c e r t a i n

Canad. J. Math. 1, 88

Science Record, Math. New Ser. 3,

613.

Assocation schemes of p a r t i a l l y balanced design = 15, and pll2 = 4. Science 2 Record, Math. New Ser. 4, 12 18. (1960).

with parameters v = 28, nl = 12, n

-

Chowla, S. and Ryser, H. J. (1950). J. Math. 2, 93

- 99.

Clatworthy, W. H. (1954).

Proc. Amer. Math. Soc. 5, 47

- 55.

P a r t i a l l y balanced incomplete block designs

with two associate c l a s s e s and two treatments per block. Nat. Bur. Standards 54, 177 (1956).

Canac

A geometrical comfiguration which i s a

p a r t i a l l y balanced design. (1955).

Combinatorial problems.

- 190.

J. Res.

Contributions on p a r t i a l l y balanced incomplete

block designs with two associate classes.

Nat. Bur. Standards.

App. Math. Ser. No. 47, Washington, D.C.

Connor, W. S. (1952). block designs.

(1958). scheme.

On t h e s t r u c t u r e of balanced incomplete

Ann. Math. S t a t i s t . 23, 57

- 71.

The uniqueness of t h e t r i a n g u l a r association

Ann. Math. S t a t i s t . 29, 262

- 266.

Connor, W. S. and Clatworthy, W. H. (1954). p a r t i a l l y balanced designs.

Some theorems f o r

Ann. Math. S t a t i s t . 25, 100

- 112.

Dowling, T. A. (1968). A characterization of cubic lattice graphs. J. Comb. Theory 5, 425 426.

-

(1969). A characterization of the Tm graph. J. Comb. Theory 6, 251

-

263. - 1Erdds, P., Renyi, A. and 568, V. T. (1966). On a problem of graph theory. Studia Sci. Math. Hungarica 1, 215 - 235. Fisher, R. A. (1940). An examination of the different .possible solutions of a problem in incomplete blocks. Ann. Eugerics 10, 52

- 75.

(1942). New cyclic solutions to problems in incomplete blocks. Ann. Eugenics 11, 290 299. Goethals, J. M. and Seidel, J. J. (1970). Strongly regular graphs derived from combinatorial designs. Canad. J. Math. 22, 449 471. Hall, M. (1967). Combinatorial theory. Blaisdell, Waltham, Mass. Hall, M. and Connor, W. S. (1953). An embedding theorem for balanced incomplete block designs. Canad. J. Math. 6, 35 - 41. Hanani, H. (1961). The existence and constmction of balanced incomplete block designs. Ann. Math. Statist. 32, 361 386. (1965). A balanced incomplete block design. Ann. Math. Statist. 36, 711. Hoffman, A. J. (1960a). On,the uniqueness of the triangular asso-

-

-

-

ciation scheme. Ann. Math. Statist. 31, 492 - 497. (1960 b). On the exceptional case in a characterization of the arcs of a complete graph.

IBM J. Res. Develop.

4, 497 -

504. Lasker, R. (1967). A characterization of cubic lattice graphs. J. Comb. Theory 3, 386 - 401. Mann, H. B. (1949). Analysis and design of experiments. Dover, New York. Masuyama, M. (1961). Calculas of blocks and a class of partially balanced incomplete block designs. Rep. Statist. Appl. Res. Un. JapanSci. Engrs. 8, 59 - 69. (1964 a). Construction of PBIB designs by fractional developent. Rep. Statist. Appl. Res. Un. Japan Sci. Engrs. 11, 47 - 54.

R. C . BOSE

.

(1964b) Linear graphs of PBIB designs. Rep. Statist. Appl. Res. Un. Japan Sci. Engrs. ll, 147 - 151. Mesner, D. M. (1967). A new family of partially balanced designs with same Latin square design properties. Ann. Math. Statist. 38, 571 581.

-

Nair, K. R. (1943).

Certain inequality relations among the combin-

atorial parameters of balanced incomplete block designs. Sankhya 6, 255

- 259.

(1950). Partially balanced incomplete block designs involving only two replications. Calcutta Statist. Assoc. W. 3, 83 86.

-

(1951 a). Statist. ~ssoc.MI. (1951 b) Statist. Assoc. Bull.

.

Some two replicate PBIB designs. Calcutta 3, 174 - 176. Some three-replicate PBIB design. Calcutta 4, 39 42.

-

Ogawa (1959). A necessary condition for existe~ceof regular and symmetric1 experimental designs of triangular type, with partially balanced incomplete blcoks. Ann. Math. Statist. 30, 1063 1071.

-

(1960). On a unified method of deriving necessary conditions for existence of symmetrically balanced incomplete block designs of certain types. Bull. Inst. Internat. Statist. 38, 43 57. Primrose, E. J. F. (1951).

euadratics in finite geometries. Proc.

Camb. Phil. SOC. 47, 299 - 304. Qvist, B. (1952). Some remarks concerning curves of the second degree in a finite plane. Ann. Acad. Sci. Fenn., No. 134, 1 27.

-

Rao, C. R. (1946). Difference sets and ccmbinatorial arrangements derivable from finite geometries. Proc. Nat. Inst. Sci. India 12, 123 - 135. (1961). A study of BIB designs with replications 11 15. Sankhya 23, 117 - 127. Ray Chaudhuri, D. K. f1962 a). Some results on quadrics in finite projective gecrmetry. Canad. J. Math. 14, 129 - 138.

-

(1962 b).

Application of the geometry of quadrics for constructing PBIB designs. Ann. Math. Statist. 33, 1175 - 1186.

R. C. BOSE

(1965). Some configurations in finite projective spaces and partially balanced incomplete block designs. Canad. J. Math. 17, 114 123.

-

Ray Chaudhuri and Wilson (1971). Solution of Kirkman's school girl problem. Combinatorics, Amer. Math. Soc, Symp. Pure Math. 19,

187 - 203. Schutzenberger, M. P. (1949). A non-existence theorem for aq, infinite family of symmetrical block designs.. Ann. Eugenics 14, 286 - 287. Segre, B. (1954).

Sulle ovali dei piani lineari finiti. R. C.

-

Acc. Naz. Lincei 17, 141 142. (1955). Ovals in a finite projective plane. Math. 7, 414 - 41.6.

Canad. J.

Seidel, J. J. (1967). Strongly regular graphs of L type and tri2 angular type. Koninkl Nederl. Akademie Van Wetenschappen - Amsterdam Proceedings series A, 70 and Indag. Math. 29, 188 - 196. Shrikhande, S. S. (1950). The impossibility of certain symmetrical balanced incomplete block designs. Ann. Math. Statist. 21, 106 111. (1952). On the dual of certain balanced incomplete block designs. Biometrics 8, 66 72. (1959 a). On a characterization of the triangular association scheme. Ann. Math. Statist. 30, 39 - 47. (1959 b). The uniqueness of the L association scheme. 2 Ann. Math. Statist. 30, 781 798. (1960). Relations between incomplete block designs.

-

-

Contributions to probability and statistics. Essays in honor of Harold Hotelling. Stanford U. Press, 388 - 395. (1961). A note on mutually orthogonal Latin squares. Sankya 23, 115

- 116.

(1962). On a two parameter family of balanced incomplete block designs. Sankhya, ser. A, 24, 33 - 40. (1965). On a class of partially balanced incomplete block designs. Ann. Math. Statist. 36, 1807

-

1814.

R. C. BOSE

87.

Shrikhande, S. S. and Raghavarao, D. (1966). A note on the nonexistence of symmetric balanced incomplete block designs. Sankhya

26, 91 88.

- 92.

Wilson, R. M. (1972 a). designs.

(A) 13, 220

89. designs.

An existence theory for pariwise balanced

I. Composition theorems and morphisms. J. Comb. Theory

- 245.

(1972 b). An existence theory for pairwise balanced 11. The structure of PBD closed sets and existence con-

jectures. J. Comb. Theory ( A ) 13, 246

- 273.

C E N T R O INTERNAZIONALE MAYEMATICO ESTIVO (C. I. M. E )

R . H. BRUCK

CONSTRUCTION P R O B L E M S IN F I N I T E P R O J E C T I V E S P A C E S

Coaso tenuto

a

Bressanone

dal

1 8 a1 2 7

Giugno

1972

Construction Problems in ~ i n i t eProjective Spaces by R H. Bruck (University of Wisconsin) Forward.

The present paper represents a complete reworking of

eight lectures given in Bressanone, Italy in the summer of 1972. The spirit of the lectures has been preserved (or s o I hope) and a l s o the order )of the material.

However, a few topics are treated more carefully here

than seemed desirable in the lectures, and a few results have been added that were unknown in the summer of 1972. The latter they were not due t o me

1.

- a t l e a s t insofar a s

- should be e a s y t o identify in the paper.

Spreads and packings of designs.

complements of I-designs.

Intrinsic construction of

In a series of unpublished lectures given in

the summer of 1963 in Saskatoon, Canada (Bruck 121) I proposed various methods of constructing finite projective planes of order n having one or more affine or projective subplanes of order not dividing n

Common t o

these methods was the idea of determining the projective plane in terms of simpler substructures.

One of the methods led me t o the concepts of

spreads and packings of 3-dimensional projective space, and these I studied and generalized during the following year in Chapel Hill, North Carolina, in close collaboration with R C. Bose and Dale Mesner.

Later,

in 1967, a t a conference in Chapel Hill (Bruck [3]) I crystalized the essence of the Saskatoon proposals in a brief preliminary s e c t i o n Now I wish t o re-examine these ideas, with the hope of focusing attention on some interesting combinatorial problems.

R. H. Bruck Let v, b, k, r,h be positive integers with

By a design D =

(y,@,I)

D consisting of a set of D, a s e t

@

with parameters v, b, k, r, h we mean a system of v (> 1) distinct objects, called varieties

of b distinct objects, called blocks of D, and an

incidence relation, I, a subset of 4) y a @

, and

X

@

. We write

21 ,

xIy, for x a

say that x i s incident with y (or y is incident with x) i f

and only if the ordered pair (xi y) i s in L We impose the following

, axioms:

(a) Each block of D is incident with exactly k distinct vari-

eties of D.

(b) Each variety of D is incident with exactly r distinct

blocks of D.

(c) If x,y are distinct varieties of D, there are exactly

h blocks of D incident with both of x, y.

Such a design may be called

The following relations are easily verified:

a A-design.

By a spread,

d , of a h-design

D we mean a collection,

d,

of blocks of D which partition the varieties of D in the sense that each variety of D i s incident with one and only one of the blocks in By a packinq,

d.

@ , of a A-design D we mean a collection,

p,

of spreads of D which partition the blocks of D; that is, each block of D i s in one and only one of the spreads in If a 1-design D possesses a spread

(1. 3) Indeed,, if

k divides v af

@. e f , then

.

contains exactly t distinct blocks, then v = kt.

In partic-

ular, i f (1. 3) holds, then a spread of D i s merely a set of t = v/k

u-

R. H. Bruck tinct blocks of D which are disjoint in the sense that no variety of D incident with more than one of them.

Furthermore, since, by the last

equation of (1.2),

a packing of D is merely a set of r disjoint sweads of D. As the literature shows, there are many applications of spreads

and (to a l e s s e r extent) of packings.

We shall be more explicit later.

For the tinie-being, we wish t o restrict attention t o spreads and packings of I-designs

(A = 1).

Let D be a 1-design

Explicitly, assume (1. I), (1.2), (1. 3)

together with

From (1.4) in the first equation of (1. 2 ) , we get

where we define n ) - 0 by

By this and (1. 3),

where k1 is a (unique) non-negative integer. of (1. I), and from (1.7),

From the second equation

R. H. Bruck b/r = r

(1- 8)

- kt =

1 t kl(k - 1)

.

If k1 = 0, then n = 0, r = 1, v = k, b = r = 1. We shall leave aside t h i s trivial case. If k1 = 1 then

If we regard the varieties and blocks of D a s points and lines, respectively, then, since

h = 1, D becomes an affine plane of order k.

It is

e a s y t o s e e that D has precisely k t 1 distinct spreads, one for every parallel class, and precisely one packing, spreads.

consisting of the k + 1

In this case, the spreads serve a s I1points a t infinity" and the

packing a s the "line a t infinity" in the usual imbedding of the affine plane of order k in a projective plane or order k. For the rest of the discussion, we assume

We notice that equations (1. 5) through (1. 8) allow u s t o express each of v, b, k, r (and A = 1) in terms of k, k1 or (as we prefer) in terms of n, k, kt.

The result of interchanging k, kt is t o give a new set of para-

meters v l , bl, k t , r l (and A t = 1).

If we a l s o u s e the symbol nl, then

and

Also, from (1.7), (1. 8), (1. 11) and (1.6),

R. H . Bruck v + b f = v f + b = 2n + n + 1 .

(1. 12)

Now let us regard the varieties and blocks of D a s the points and lines, respectively, of a partial plane n o , (with incidence given by the incidence relation of D). We ask under what conditions no can be imbedded in a projective plane n subject t o the following requirement: (*)

The only lines of n incident with a point P of

To

are

the lines of no incident (in no) with P. If such a projective plane n exists, then we know its order. Indeed, every point P of

%

+1

i s incident with precisely r = n

lines of

T;

hence n has order n, Moreover, since no has v distinct'points, b distinct lines, whereas n has n2

+n+1

distinct points and a like

number of distinct lines, then, by (1. 12), the numbers of distinct lines and points (note the order!) of n which are not in no are v f and b' respectively.

Consider a point Q of n which is not no.

of n i s incident with Q and also with a point of must be a line of 'rrg. points of 'rrg.

T,,,

If a line L

then, by (*), L

Therefore L i s incident with exactly k distinct

Consequently, Q must be incident (in n) with precisely

distinct lines of 'rrg - and, clearly, these lines form a spread of 'rrg (or D).

Since Q i s incident in

then Q must be incident in which are not in no.

T

T

with precisely n

+1

Next let M be a line of

.,

T

.

+ 1= r

T,

with precisely kf distinct lines of .rr which i s not in no.

Then, by (*), no point of 'rrg is incident (in n) with M. incident in n with n

lines of

distinct points of

T

Hence M i s

which are not in roo

R. H. Bruck Furthermore, the r spreads of no defined by the r points of M partition t h e l i n e s of 'rrg

- since every line of

and hence form a packing of

'rrg must l1rneetlt M i n a point

-

%.

Now (assuming that n e x i s t s subject t o (*)) let D1 be the system of which t h e varieties are t h e l i n e s of x not in 'rrg, t h e blocks are t h e points of n not i n n ,, incidence relation of n.

and t h e incidence relation is t h a t induced by t h e Then, by t h e above discussion, D1 is a design

with parameters v l , bl, k t , r t , A t (with r1 = r, A t = 1) where v t , b t , are defined by (1. 11). W e shall c a l l such a design D1 a complement of D. Note that we d o not accept every design with t h e correct parameters a s a complement of D.

The d e s i g n D and complement Dl, by definition,

fit together a s indicated above t o define a projective plane of order n Note further that, if n e x i s t s subject t o (*), then two disjoint blocks B, C of D, considered a s l i n e s of 'rrg, must b e incident in n with a unique point Q of n not in no, and hence must b e contained in the unique spread of D defined by Q. With t h i s a s a clue, we c a n s e e how t o construct a complement D1 of D (if one e x i s t s ) in terms of spreads and packings of D. W e need the existence of a collection, a collection, (1) (11)

ep,of packings of

gs

of spreads of D and

D subject t o the following requirements:

Each packing i n &P is made up of spreads in gs. If B, C are two disjoint blocks of D, there e x i s t s one

gs which contains B and C. If 4,d2are two disjoint spreads in & s, there (111) which contains dl and d2. exists one and only one packing in P and only one spread in

Now let D1 be t h e system of which the varieties are the packings in

P'

the blocks are t h e spreads i n

cs,and t h e incidence relation

R. H. Bruck is the containing relation, is incident with

then

If

g,

$

iff

A

are in '

gp, Es respectively,

is contained i n

@

. W e claim

that Dl is a design and a complement of D. To begin with, we need t o show that the cardinal numbers of p,

tsare given by

The proof of (1. 13) will show somewhat more. Consider a block B of D.

W e c a n choose a treatment x of D

which i s not incident with B in

ways (cf. (1.7). (1.6)).

If y is a treatment incident with B, there is a

unique block B1 of D incident with x and y; a s y ranges over the k treatments incident with B, B

ranges over k distinct blocks. Hence 1 the number of distinct blocks of D which are incident with x and disjoint from B is (cf. (1. 11))

Therefore the number of ordered pairs x, C such that x is a treatment and C is a block of D, x is incident with C but not with B, and C is disjoint from B is

(cf. (1.6)). (1. 14)

For given C, x can be chosen in k ways.

Hence

For each block B of D, there are exactly (k distinct blocks of D disjoint from B

.

- l)vl

R . H. Bruck

Now we use (11). If C is a block of D disjoint from B, there is a unique spread in

&,

containing B, C, and

Q !

contains exactly

(cf. (1.7))

distinct blocks disjoint from B.

By this and (1. 14):

Each block of D is contained in exactly

(1. 15)

vl/kl = vl/r distinct spreads of D in

gs .

Since D has b distinct blocks,and since each spread consists of v/k = b/r

(disjoint blocks, we s e e from (1. 14) that

This proves the second equation of (1. 13). Next consider a spread ef in by (1. 15), the number of distinct spreads but are distinct from

Each such

$

blocks B in which meet

Since

and

d

$

$

in

gs

d.

which contain B

is (cf. (1. 1))

intersects is v/k.

s. For each block B in

d

in B, by (11).

The number of distinct

Hence the number of distinct spreads in

in a block is

gs

R. H. Bruck

then bl

- (1 + v(kt - 1)) = r - 1 + kkt(k' - 1) = n

+ n(kt - 1) = nk'

.

Therefore

c s i s disjoint from exactly nkl distinct spreads i n zs. Next we u s e (111). Given a spread d in rS, we c a n choose a 8 in disjoint from i n exactly nkl ways, by (1. 17). Each spread in

(1. 17)

spread By (111).

$, 8' a r e contained

number of distinct spreads in

is r

- 1= n

in

b unique

packing,

a), in kp.

The

@ distinct from (hence disjoint from) $(

Hence

(1. 18)

&

Each spread in

is contained in exactly

k t distinct packings i n Since, by (1. 18),

we have completed the proof of (1. 13). W e have a little more t o do. Then

@

c o n s i s t s of r.,'=. n

&

(I), each of t h e s e is in (1.18), exactly k1 are distinct from

-

@.

+1

Consider a packing

in

$

P'

distinct and disjoint spreads of D.

p, there

For e a c h spread df of

1 distinct packings in If

@)

P

6) is such a packing,

which contain

@

n

=

By

are, by

d

and

by (111).

R. H. Bruck Hence t h e number of distinct packings in

&

P

which intersect

@

in a

(single) spread is

I n other words: Every two distinct packings i n

(1. 19)

have a unique common spread i n

P S

.

In view of (1. 13), (1. 19), (1. 18), we see that Dl, a s defined above in terms of v l , b', kl,r,A

& s, Gp,

(A = 1).

is a design with parameters

Now we define n a s a system whose points are

t h e treatments of D and t h e blocks of Dl and whose lines are t h e blocks of D and t h e treatments of Dl. Within D or Dl, the incidence relation of n is that of D or Dl, respectively. a block of Dl (that is, a spread of D in by t h e containing r e l a t i o n no packing of D in

Incidence in n between

and a block of D i s defined

Finally, by f i a t , no treatment of Dl (that is,

E p ) is incident in n with a treatment of D.

n is a projective plane of order n, and Dl is a complement of D. By symmetry, D is a complement of Dl. I n t h e next few sections, we shall d i s c u s s I-designs D with parameters of form (1. 1) - (1. 9), for special values of k.

Then

R. H. Bruck 2. ovals.

(k = 2. ) Cliques.

Complete symmetric graphs.

Complete

Let D be a 1-design corresponding t o the situation of section 1

i n which

where m i s a n integer. (2.2)

Then the parameters of D are given by

v = 2 m + 2 , b = ( m t 1)(2m+l), k = 2 , r = 2 m + 1 , X = 1.

1

Also

Furthermore, the parameters of a complement Dl of D (if one exists) are given by

The design D is variously called a complete symmetric graph (or a clique) with v vertices, a complete v-point or a complete (or completed) oval.

In this special case, the concept of a complement seems t o have

been introduced by Esther Seiden [13], and I was aware of some of her work before formulating the material of $1.

8 , of a projective plane s e t of n + 1 distinct points of

Let us review the concept of a n oval, of finite order n. -rr

By definition,

is a

no three of which lie on a line of .rr.

tangent or non-secant t o tinct points of

&f

We call a line L of

.~r a

.rr

secant,

according a s L contains 2, 1 or 0 dis-

@. Every point of w l i e s on n distinct secants and a

unique tangent. If n i s odd, the points of -rr not on

fl form two c l a s s e s ,

the

R. H. Bruck interior points of

fl

@.

and t h e exterior points pf

An interior point

lies on (n + 1)/2 distinct secants, 0 tangents, and' (n + 1)/2 distinct non-secants.

An exterior point l i e s on (n

distinct tangents, and (n

- 1)/2

2 distinct interior points, (n

- 1)/2

distinct secants, 2

distinct secants.

+ n)/2

There are (n2= -ii)/2

distinct exterior points, and the n

Ct

points of

If n is even; assume (2. 3).

The points of

8

not on

?T

two c l a s s e s , one consisting of a single point, K, called the The point K l i e s 04 each of the n line of

+1

?T.

+ 1 = 2m + 1

Each of the remaining n2

form

&

of

tangents and on no other

- 1 = 4m2 - 1

#

points not on

l i e s on m distinct secants, a single &angent]line through K, and m non-secants. We complete n

+ 2 = 2m + 2

to

by adjoining K.

Then

&* consists of

distinct points, no three collinear.

The lines of

two c l a s s e s with respect t o

B*, since each line of

of

@.

* @

or a non-secant of

lies on exactly m + 1 secants of

Each point of

* @

?T

.rr

form

?T

i s either a secant

&*

which is not on

and m non-secants of

4

9

@, together with its secants, is the design D. * lines of C?' , together with the points of n not on

. The

complete oval

The

non-secant

&,

*

form a complementary design Dl. It should now be e a s y t o check that, for D with parameters (2.2), the existence of a complement is equivalent t o the existence of a projective plane of order n = 2m which possesses a n oval (and thus a complete oval. ) The question a s t o whether every finite projective plane (of odd or even order).has a n oval seems t o have remained unanswered.

There are

non-isomorphic projective planes of the same order each having ovals, and there are many examples of a projective plane having differently embedded

R. H. Bruck

ovals.

Thus, i n the present context, D might have two non-isomorphic

complements, but we could not decide without further examination whether the corresponding projective planes were isomorphic or not.

Our present intent is t o start from D and try t o construct a complement, Dl.

First, however, it is useful t o begin with some complete

ovals occurring in a projective plane over a field F = GF(n) where

W e choose' integers s, t of form

This ensures that the mappings

are automorphisms of F = GF(n) distinct from the identity, and that each is the inverse of the other.

We a l s o want the mappings

t o be one-to-one mappings of F upon F.

The conditions for this t o be

true are ( n - 1 , s - 1) = l = ( n - l , t

-

1)

or, in view of (2.7), (2.9)

--fe,f) = 1

.

Finally, t o avoid duplications, we impose the equivalent conditions

R. H. Bruck Now l e t us work'in the affine plane A over F = GF(n) and consider the affine curve

Here

$(s)

c ( s ) defined by either of the equivalent equations

consists of t h e n distinct affine points (x,y) of form

(k, ks) where k ranges over F.

For any fixed k, c c F with c

#

0, the

line

meets

g(s)

where u

F

in the point (k, ks) and in a second point (k + u, kS

F, u

#

+ us)

0 is determined by

The lines y = c, in the parallel c l a s s of the x-axis, and the lines x = c, i n t h e parallel c l a s s of the y-axis, each meet Hence, if

* (s)

g ( s ) i n a unique point.

is the point-set of the corresponding projective plane

defined by &*(s) =

(2. 12)

Z(S)u {x,Y)

where

X = { Y = c / c c F) ,

are t h e llpoints a t infinity" on the x-axis and y-axis respectively, then $*(s)

is a complete oval of the projective plane over GF(n). It may be shown that, regardless of the (admissible) choice of s , t,

every collineation of the projective plane which maps must fix Y.

Thus it is natural t o think of

* (s)

upon itself

R. H. Bruck

g (s, u {XI a s the oval and of Y a s the knot. X and

The group of collineations which fix

G ( s ) is easily specified.

sub-group (on the points of

There is a sharply doubly-transitive

c ( s ) ) consisting of a l l the mappings

and the full group is generated by the group (2.13) and the mapping

When t = 2 (and not otherwise) there i s a l s o a collineation which moves X, namely (2. 15)

(For t = 2) : (0,O)

(x,y)

-

-,X

X

(0,O)

,

#o

.

Y--Y, (x-l,x-'y)

for x

Consequently, for t = 2, the group generated by the group (2. 13) together with the mappings (2. 15) is sharply triply transitive on For e <_- 4, the above conditions require t = 2.

r ( s )u

{XI.

However, for

e = 5 we can have both

and

Thus, for e _> - 5, we can expect differently imbedded complete ovals. There are a l s o the ovals of Tits, Luneburg and Suzuki, but we will not go into them here.

See Luneburg [lo].

n. n. 3.

(k = 2 continued. ) Construction of complements.

Drucn

As in sec-

tion 2, D is a design with parameters (3.1)

v = 2 m t 2 , b=(mt1)(2m+l), k = 2 , r = 2 m + l , A=1.

If there exists a complement, Dl, of D, then Dl has parameters

and the projective plane defined by D, Dl has order

Here m is a positive integer and, t o avoid trivialities, we assume m > 1. We may assume that the varieties, or points, of D are the first v = 2m

+2

positive integers.

The blocks (or lines), the spreads and

packings of D may be expressed in terms of elements of the symmetric group on these points a s follows: A line is a transposition,

A spread is a n

involution without fixed points; that is, a product of v/2 = m + 1 disjoint transpositions.

A packing is a s e t of r = 2m

+1

distinct involutions with-

out fixed points, such that the product of any two distinct involutions of the s e t a l s o has no fixed points. The numbers, s(m), of distinct spreads of D can be written down a t once:

We notice that, for m _> - 2,

Since, in constructing Dl, we use just b1 (suitably chosen) spreads, we

R. H. Bruck s e e that for m large we need only a small fraction of the spreads.

But

for m = 2, we need a l l of them. It might be interesting to have a formula for the number, p(m), of distinct packings of D.

As we shall see,

Now we shall consider the details for small values of m, beginning with the excluded c a s e m = 1. I.

m = I.

Here

We think of the four points 1,2,3,4 quadrangle.

a s vertices of a planar

There are 3 spreads, namely

These represent the diagonal points of the quadrangle. unique packing.

They a l s o form the

The latter represents the line joining the diagonal points

in a projective plane of order 2. 11.

m = 2.

Here

And, a s pointed out, the total number of spreads is s(2) = 5! ! = 15. To find the spreads and packings, we may divide the six points

R. H. Bruck

1,2,3,4,5,6

into two.unordered triangles, say { 1,2,3)

and { 4,5,6).

If a spread contains a line of one triangle, then it contains one line of each triangle and a single "cross-joinff, a line joining the remaining points of the two triangles.

There are 9 such spreads, a typical one being

Note that 8 i s uniquely determined by its single cross-join, (36). The remaining 6 spreads each consist of 3 distinct cross-joins; and there are 2 such spreads for each cross-join

Thus there are just 4 spreads of this

second type disjoint from 8. Arranged in terms of the cross-join containing the point 3, these are:

%= We note that

(16)(24)(35) ,

,+2

i, j = 1,2, '#'i and

+.J

.

$2 = (14)(26)(35)

are disjoint and q1 ,q2 are disjoint but, for have a common line.

Now we consider a packing. spreads which partition the lines.

This must consist of 5 disjoint The spreads containing the lines (12),

(13), (23) of the first triangle must be of the first type and must be three disjoint spreads.

- single cross-join

The remaining two spreads cannot

contain a line of triangle { 1,2,3) , and hence must be of the second type. In particular, a packing containing 9 must either contain

qj2 or con-

tain Jll,q2, together with two more spreads of the first type.

There are i n

fact exactly two packings containing 9, namely

and

-

R. H. Bruck

and these have only 8 in common If we note, in addition, that the four newly displayed spreads are the only ones of the first type disjoint from 8, we will have that t o each spread a disjoint from 0 there corresponds one and only one packing containing 8 and a.

Since every spread i s a

conjugate of 8 i n the symmetric group on 1,2,3,4,5,6,

we have com-

pleted our investigation We may use the above work t o show the uniqueness of the projective plane of order 4.

First we verify, by a counting argument, that such a

plane must contain a s e t of six distinct points, no three collinear.

Then,

by the above analysis, the plane is uniquely determined by the s i x points. 111.

m = 3.

Here

Also s(3) = 105. We know from the beginning that D has no complement Dl, since there i s no projective plane of order 6.

Nevertheless the spreads and

packings of D are worth studying, both in themselves and for use with larger values of m First we shall consider a type of packing which may be called projective.

Behind this concept is the fact that, in many ways, the 8

points of D can be put into one-to-one correspondence with the 8 points of affine 3-space over GF(2). Once such a correspondence has been chosen, the lines of D fall into 7 disjoint parallel c l a s s e s of 4 lines

R. H. Bruck each

Each such parallel c l a s s is uniquely 'specified by a spread of D.

The 7 spreads form a packing which represents the plane a t infinity, with the 7 spreads a s i t s points.

Finally, the 7 spreads fall into 7 distinct

partial packings of 3 spreads each which represent the 7 lines a t infinity. Let u s remark before going on that a similar possibility arises in c a s e k = 2, v = 2e, e > 2.

In t h i s case a projective packirg would repre-

sent, not a plane, but the hyperplane a t infinity of a n affine e-space over GF(2). I t would uniquely determine projective e-space over GF(2), since it would specify a l l the parallel c l a s s e s of lines of affine e-space.

An intrinsic definition, adequate for the present case (v = 8), may be given a s follows: A packing,

@ , of

D is called projective provided

that, for any four distinct points a, b, c, d of D, if there is a spread 8 in

@ containing the lines

(ab), (cd), then there is a spread

containing the lines (ac), (bd).

+

in

@

(Thus also, on interchange of c, d, there

+

in

@

definition: If

@

represents the plane a t infinity of affine 3-space over

is a spread

containing the lines (ad), (bc). ) Reason for the

GF(2), then 8 represents a parallel class of lines and hence a, b, c,d are the four points of an affine plane (of order 2. ) We note the following: (i) Each spread, 8, of D is contained in exactly two distinct projective packings.

The two packings have only 8 in common

To s e e this, consider the spread

Applying .the criterion t o the four points 1 , 2 , 3 , 4 and a l s o t o the four points 5,.6,7,8, we s e e that a projective packing containing spreads of form

e must contain

R. H . Bruck

where a , p are (57)(68), (58) (67) in one of the two orders.

We shall go

on with the c a s e that

If the spread is

In any packing there must be a spread A containing (15). projective and contains 8,

+ ,+,

must contain (26), (37). (48).

then, by reference t o 8,

+, +

in turn, A

Hence

Continuing in t h i s way we arrive a t the following packing:

We need t o verify that case, and

Q

@

is projective, but t h i s is indeed the

is the unique projective packing containing

second projective packing;

'@I,

containing

fashion or, alternatively, by transforming Clearly

(3

and

(?I

(?

e

The

8, +,$.

may be obtained in similar

by the permutation (78).

have only -9 in common This proves (i).

Next:

R. H. Bruck (ii) There are excictly 30 distinct projective packings of D. Theke form a single orbit under the symmetric group S8.

If

is a pro-

jective packing and G(@ ) is the subgroup of S8 mapping (P upon then G((?)

is sharply transitive on ordered tetrahedra.

@,

In particular,

To begin with, we know that each of the 105 spreads is contained in exactly two distinct projective packings.

On the other hand, each

Hence there are exactly

packing contains exactly 7 spreads.

distinct projective spreads. Let 8 be a spread and let fpackings containing 8. Let packing

2.

+

(?,

be the two distinct projective

be a spread contained in a projective

There exists a permutation A in S8 such that

consequently,

@

1 - l =~ ~ or

If h- 1Qh =

6

( 3 1 .

then, by the proof of (i), there exists a 2-cycle

r in S8

such that T

Hence

2

A

is a conjugate of

a conjugate c l a s s If

-1 -1

@

Qh-r=@.

p.

Thus the

- a single orbit under

30 projective packings form

S8.

is a projective packing, there are several ways of determin-

R. H. Bruck

ing G(@). One i s a s follows: GF(2), and

(P

makes D into affine 3-space over

G(O) i s the corresponding

group of collineations.

We can

pick an ordered triple a, b, c of distinct points of D in 8 7 6 distinct

@)

ways.

The plane of a, b , c (with respect t o

contains a unique fourth

point.

Hence we can pick a point d, not in this plane, in 4 ways.

That

is, the number of ordered tetrahedra

of D i s 8 a 7 6 * 4.

Given such a tetrahedron, the remaining four points of

D lie one each on the four faces of the tetrahedron

Hence, clearly,

G((P) i s sharply transitive on ordered tetrahedra and has the stated order. This proves (ii). We could also calculate

IG(@)

( a s follows: Since the

30 projec-

tive spreads form a single orbit under S8 then

and hence

Before we introduce the next type of packing, we shall investigate the orbits of the spreads of D under conjugation by the cyclic group ( T

)

generated by

The 105 spreads break up into 15 orbits of length 7.

A straightforward

calculation shows that exactly three of these orbits are packings, namely:

R. H. Bruck

@,

: orbit of (12)(35)(48)(67)under trr ) ;

p2: orbit of (12)(37)(45)(68) under

2

(T

);

: orbit of (12) (38)(47)(56) under (n)

The packings

.

PI, O2 are easily verified t o be projective.

The remain-

ing packing,

i s not projective. Let a be the unique element of S8 which fixes 1,2 and satisfies

- l 7 r O = a3 .

a Thus

a= ( 3 5 4 8 6 7 ) .

Clearly u must permute the packings

and

Pl, p2,

In fact

R. H. Bruck

Furthermore, o fixes the first spread of

2

six spreads of

2

'and permutes the remaining

in a single cycle.

It i s easy t o see that the only permutation in G ( 3 ) which fixes 1, 2 and 3 i s the identity.

Again, if ~ ( 2 contains ) an element which

moves 1, then, in view of the properties of triply transitive on D.

In particular, G(%)

R,

must contain a cycle of

length six which fixes 3, 4 and maps 1 upon 2. ends in a contradiction. (iii) G(%)

a, G ( 2 ) must be sharply A straightforward search

Hence:

i s a non-abelian qrouD of order 42 such that

GR)fixes 1 and is sharply doubly-transitive

(iii)(a)

on the

remaining 7 elements of D and (iii)(b) ~ ( 3 i)s sharply doubly-transitive on the 7 spreads of

a We shall call any packing equivalent t o

2

a non-projective

cyclic packing. (iv) Let the 7 spreads of

&! @.

be a packinq of D such that G(&) is transitive on Then

&!

is either a projective packing or a non-

projective cyclic packing. Proof. -

Since G = G(&) i s transitive on the spreads of

where n i s the order of the subgroup of G fixing a spread of the centralizer in S8 of a spread has order 2'

&?,

Q0 Since

I!,

A Sylow 7-group of G has order P a n d i s a subgroup of S8.

Hence, after

R. H. Bruck replacing

@

by a conjugate, we may assume that G has a Sylow

7-group generated by the perhutation

(P,

must be

or&

T

(see above).

In this case,

@

This proves (iv). We may verify further that

(v) There are exactly 960 distinct non-projective cyclic packings of D, and these form a single orbit under S8.

Proof. I t is enough t o show that S8 has exactly 960 distinct Sylow 7-subgroups.

Such a subgroup fixes a unique point a of D; and

a can be chosen in 8 ways.

The number of 7-cycles fixing a is 6!, and

these lie in s e t s of 6 in the Sylow 7-subgroups fixing a.

Hence there are

precisely

Sylow 7-subgroups.

This proves (v).

To make further progress, we need the following: (vi) If

8 is a spread of D, there are precisely 60 distinct

spreads of D disjoint from 8.

These split into two orbits, of lengths 12

and 48, under conjuqation by the centralizer of 8 in S8.

The orbit of

length 12 consists of a l l spreads of D which are disjoint from -9 and commute with 8; t h i s orbit a l s o consists of

C/>. @ '

u

-

{ 8 ) where

a r e the two regular packings of D which contain 8. If h is in

t h e orbit of length 48, then 8A is the product of two disjoint cycles of length 4, and

(m)'

is the unique spread of D which is disjoint from and

commutes with each of @,A. Proof. -

To begin with, for 0 <_- i 5- 4, let f(i) be the number of

R. H. Bruck spreads of D which are distinct from 8 and have exactly i lines in common with 8. It should be obvious that

Next we calculate f(2). We can choose two distinct lines of

8 in 6 ways.

Once t h e s e a r e chosen, we can construct a spread having precisely these 'two lines in common with 8 in 2 ways.

Hence

Next we calculate f(1). We can choose a line of 8 in 4 ways.

The

number of spreads of D having only this line in common with 8 is

Hence f(1) = 32, f(0) = 6 0 . Thus there are precisely f(0) = 60 spreads of D disjoint from

8.

Next let @ be a spread disjoint from 8 which commutes with 8. Then, under conjugation, @ must permute the 4 lines of 8 in two cycles of length 2.

u

-

Equivalently, @ must be one of the 12 spreads in { 8).

From previous work, or by direct calculation, the latter

s e t forms a single orbit under the centralizer of 8 in S8. For the last part of the proof, we may assume that

R. H. Bruck

The spread

is disjoint from

8.

Also

Clearly (8ho)2 i s a spread which is.disjoint from @,Ao and commutes with both of them.

The common centralizer of @,Ao is easily s e e n t o be

sharply transitive on the elements of D.

Thus i t h a s order 8.

Indeed, it

c o n s i s t s of the identity permutation, the 5 spreads (rn0l2 and

and t h e elements (1485)(2376) , (1 584)(2673) of order 4. from

Clearly the only spread in t h i s common centralizer disjoint

8,A0 is (@ho)LThe centralizer of 8 h a s order

Hence t h e number of conjugates of A.

This is enough for the proof of (vi).

under t h i s centralizer is

R. H. Bruck

In addition we need: (vii) Let

@, (?

be the two distinct projective packings contain-

ing a spread 8 of D. Then each spread a i n from exactly 4 spreads in If

@-

{ 8)

is disjoint

- { 0) , but commutes with none of these.

B is any one of the 4 spreads i n

@' - { 8)

disjoint from a , then

(aB12 = 8. Proof. -

The first two sentences of (vii) require a straightforward

verification which we will omit.

The l a s t sentence then follows from (vi).

Now we come to: (viii) Let S be a commutative s e t of two or more disjoint spreads of D.

Then there e x i s t s one and only one projective packing, rS1, of D,

which contains S. Proof.

If a , p are two distinct (hence disjoint) elements of S,

then, since a g = pa, there exists a unique projective packing containing

a , @ . C a l l t h i s [a,p]. If IS ( = 2, there i s nothing more t o prove. If S h a s third element Y distinct (and disjoint) from a ,p , then there exists a unique projective packing [ a , Y] containing a , Y. [a, p]

#

If

[ a , Y 1, then [ a , p], [ a , Y ] are the two distinct projective pack-

ings containing a.

Hence, by (vii), pV

#

Yp, a contradiction.

Therefore

(and similarly)

If

( SI = 3, we are done.

from a , p , Y, then

If S has a fourth element 6 , distinct

R. H. Bruck

[a,Bl =.[B,Yl

= [Y,61

This shows that every two distinct elements of S are contained i n the same (unique) projective packing, which we may c a l l [S]. Thus (viii) is true. (ix) Let S be a commutative s e t of 6 disjoint spreads of D.

Then

there is one and only one packing of D containing S, namely the projective packing rS1. Proof.

Let

8 be t h e unique element of [S] which i s not in S.

If a packing contains S and a seventh spread $, then 8, $ cover the same lines of D, s o 9 = 8.

This proves (ix).

(x) Let S be a commutative set of 5 disjoint spreads of D.

Then

there is exactly one packing of D which contains S but is not projective. The s e t of a l l packings of D which contain a commutative set of 5 disjoint spreads but are not projective consists of precisely 630 distinct spreads and forms a single orbit under conjugation by S8. Proof. not i n S.

Let

8, 9 be the two distinct spreads which are in [S] but

In view of (vi), after replacing S by a suitable conjugate, we

may assume without l o s s of generality that

We wish t o form a newpacking by replacing 8', $ ' which cover the same lines a s 8,$. contain (12), (13) respectively. cannot contain (24). Therefore

8, $ i n [S] by two new spreads We may suppose that O f ,

Then $I cannot contain (34) and

8'

$I

R . H. Bruck

where 1 , ~ are disjoint and each contain two disjoint lines chosen from

Clearly 1 , ~ must be

in some order.

If A = (56)(78), we get 8' = 8,

$I

= $, contrary t o desire.

Hence we must have

and

The packing obtained by replacing 8, $ in [S] by

el,

$ ' is not

projective, since it does not contain 8, $, whereas the projective packing [S] i s determined by any two distinct spreads in S.

Note, also, that of

the two projective packings of D containing the element 8$ of S, one i s [S] and the other contains

el, $'.

Hence each of O f , $ ' commutes

with exactly one element of S, namely 8$. Conversely, let 8, $I be the spreads displayed in the preceding paragraph, s o that

[e, $1

i s the projective packing

proof of (i), Then S is the set of spreads of and we are interested in the packing 8, $I by

el, $I1.

&?

ip

@

displayed in the

distinct from 8, $I,

obtained from

@

by replacing

Clearly G($?) is isomorphic t o that subgroup of

~(0

R. H. Bruck Hence, by (ii), G(@) is sharply which fixes the unordered pair 8, $J. transitive on those ordered'tetrahedra

of D (with respect t o @ ) such that one of the lines (ab), (ac) i s in 9 and the other i s in 9. Therefore

and the number of conjugates of

dZ

under S8 is

IS8: ~ ( 6 ? ) 1 =7 * 6 * 5 * 3 = 6 3 0 . This completes the proof of (x). Next we need two simple concepts.

Let

{ a , f3 ,Y ) be a commu-

tative s e t of three disjoint spreads and let

be the unique projective packing containing a , p , Y. (projective) plane a t infinity, and the spreads of

Then

@

infinity, of the affine 3-space of D determined by

Q

is the

are the points a t

p.

I t may be verified

that the line a t infinity containing a,@consists of the three disjoint spreads a, 3 ! and a@=

m.

i s a line a t infinity if

Hence we make the following definitions:

af3 = Y and is a triangle a t infinity if af3 # Y.

(xi) There are precisely 420 distinct packinqs

&

{ a , f3, Y

@

of D such that

contains a maximal commutative subset S of 4 disjoint spreads of D.

These packings form a single orbit under conjugation by S8. is such a packing,

Thus, if

&

R. H. Bruck

(G(@)I=8 . 4 -

3.

Moreover. if S is a commutative s e t of 4 disjoint s ~ r e a d sof D, a necessary and sufficient condition that S be imbeddable a s a maximal commutative subset of a packing

dij

is that the complement of S in the

projective packing [S] be a line a t infinity. isfied,

@

When the condition is sat-

can be chosen in two distinct ways; the two packings have

only S in common .Proof.

Let S be a commutative set of 4 disjoint packfngs.

assume without l o s s of generality that [S] =

@

We can

where p i s the projective

packing exhibited in the proof of (i). Let K be the complement of S in

P. Case 1. K is a triangle a t infinity.

Then, by (ii), we can assume

without l o s s of generality that K consists of the disjoint spreads

If S can be imbedded a s a maximal commutative subset in a packing then

I$?

must be obtained from

@

by replacing

8, +, X by three dis-

joint spreads

which cover the same lines a s

&?,

8 , + , X but are distinct from 8,

+, X.

Clearly one of the lines (24), (26) must be in 9' and the other in X.

R. H. Bruck There are two cases:

Clearly the line (57) must not be in cannot be in X in

el,

XI,

I.

$I3

(else we will have 9 ' = 9 ) and

Therefore (57) must be in 8: Similarly, (37) cannot be

e l s e X = X , and cannot be in 9'.

a contradiction

Hence Case 1. 1 cannot occur.

'The line (34) cannot be in 9' or cannot be in 8' or

This puts both (57), (37) in

+I,

XI,

hence must be i n

hence must be in X

I.

C a s e 2.

8'.

But then

Hence C a s e 1.2 and Case 1 are impossible.

K is not a triangle a t infinity.

Then (37)

This gives

Thus (48) must be in 9 ' and hence (78) must be in

a contradiction.

8'.

That is:

This leaves:

K is a line a t infinity.

of generality that K consists of

Then we may assume without l o s s

R. H. Bruck

We s e e easily that three disjoint spreads covering the same lines a s 8, @, J, must be made up by pairing the three products

,with the three products

in some order.

We may regard

8 , 9 , J, a s giving the normal pairing.

Then, if the new spreads are t o be distinct from 8, $,+ we s e e that these are just two solutions, corresponding t o the two derangements of the normal pairing: Either we can replace 8, @, $ in

getting a packing

d?

getting a packing

a2-

In each of

or we can replace

(Rl, : z 2 , S

@

by

8, @,

+

in

by

occurs a s a maximal commutative s u b s e t

Let H be the normalizer in S8 of the subset

R. H. Bruck K = {8,9,11,) Since [K] =

p, H

.

is a subgroup of G@).

Hence, by (ii), it is clear

that H is sharply transitive on the ordered tetrahedra

of D (relative t o of 8,9,$.

@

) such that the lines (ab), (ac) belong t o some two

Thus

Clearly each element of H either normalizes both of the s e t s

or interchange the two sets.

The permutation

is in H and interchanges t h e two s e t r

Consequently

conjugate under H (and S8!) and

is a subgroup of index 2 i n H, Thus

and IS,: G ( ~ % ? ~ )7I = * 6 * 5 * 2 =4 2 0 . This completes the proof of (xi).

R1,g2 are

R . H. Bruck

The following statement throws additional light on the packings described in (x) and (xi). (xii) $et A be a commutative s e t of 4 distinct spreads of D such that the complement. L. of A in the projective packins [A1 is a line a t infinity.

Then a line at infinity. T, can be chosen i n precisely 5 distinct

ways such that T is disjoint from A and the set

6= T u A

ing which contains T a s a maximal commutative subset.

is a pack-

In two cases. T

i s disjoint from L and A is a maximal commutative subset of

&

.

(Com-

pare (xi). ) In the remaining three cases, T meets L in a spread, say a, and S = A u ( a) is a maximal commutative subset of Sketch of proof.

&.

(Compare (x).l

If T is t o be disjoint from L, we consider the

proof of (xi) with S = A. If T is t o meet L in a spread a (which can be chosen in 3 ways) we consider the proof of (x) with S = A u { a). The following gives still another look a t (x): (xiii) Let a.0

be disjoint spreads of D such that a0

.-

# Ba.

Let L. M be the unique intersecting lines a t infinity which contain a. B respectively.

Thus

Then there are exactly two distinct packings of D which contain L u M. In one of these, L is a maximal commutative subset and M is contaiced in a maximal commutative subset of 5 distinct spreads.

In the other the

rol'es of L. M are interchanged. Proof.

We may assume without l o s s of generality that

R . H. Bruck

Then we find that

Thus the 5 spreads in L U M are disjoint but do not cover the lines

Hence the remaining disjoint spreads Y, 6 of a packing containing L u M must pair the products

with the products

This gives the two packings

where

It i s easy to check that, in

R1,M

and 1 u { Y1,

are maximal com-

R. H. Bruck

mutative subsets and, in mutative subsets.

a2,L and

M u

Y2, 62)

are maximal com-

This proves (xiii).

Next we shall need the. following: (xiv) Let a, p be disjoint spreads of D such that a p # @. Then there are precisely 31 distinct spreads of D disjoint from both of

a, p.

Of these: (a) 1 (namely (ap)2) commutes with both of a, f5; (b) 5 commute with a but not with p; (c) 5 commute with p but not with a ; (d) 20 commute with neither of a, p. Sketch of proof.

A s in the proof of (vi), for 0 <_- i

<- 4,

let f(i)

be the number of spreads of D which are distinct from a and have exactly i lines in common with a.

In addition, for 0 5 - i, j <_- 4, let

g(i, j) be the number of spreads of D which are distinct from have exactly i, j lines in common with a, p respectively.

a, p and

By the proof

of (vi), and by symmetry,

and

Next we specialize a, p ( a s warranted by (vi)) and verify that

R. H. Bruck

From these values in the above equations,

and

This proves the first statement of (xiv). To prove (a), (b) we need merely re-examine the 12 spreads disjoint from a and commuting with a in relation to p.

(Compare (vi), (vii). ) And (c), (d) follow from (a), (b).

This proves (xiv). We shall call a set S of n disjoint spreads of D anti-commutative if either n = -

1 or n

>

1 and

a@# pu for every two distinct spreads

a, f3 in S. (xv) Let S be an anti-commutative set of n disjoint spreads. Then the number, N(n), of non-projective cyclic packings of D which contain S, i s given a s follows:

However, N(5) = 1 or 0, dependinq on the case. Sketch of proof. packing

We begin by noting that the non-projective cyclic

2 , displayed after the proof of (iv), i s an anti-commutative

of 7 disjoint spreads.

From this and (iv), (vi), we have

set

R. H. Bruck

whence N(l) = 64, N(2) = 8.

Next we note that the first two spreads of

2

are the disjoint

pair

We need, in effect, t o examine the 8 cyclic packings containing a, @ and, for this, the following considerations prove useful.

Since a@

#

fa, it

,

turns out that for each ordered pair d , d l of elements of D (not necessarily distinct) there is one and only one permutation of D which interchanges

a, p (by conjugation) and maps d upon dl. In particular, the eight permutations interchanging a, @ are the four odd powers of

together with 7 = (23)(48)(57), fixing 1,6 ;

p = (14)(37)(68), fixing 2,5 ;

v = (16)(27)(35), fixing 4,8 ;

p = (18)(25)(46), fixing 7 , 3

.

Clearly, under conjugation, A induces the cycle

Each cyclic packing containing

a,f3 must be the orbit of a

under successive conjugation by a permutation of D which fixes one

R . H. Bruck

element of D, permutes the remaining elements of D in a cycle of length 7, and maps a upon p.

However (as may be verified) there are, in all,

8 distinct cycles of length 7 which fix 1 and map a upon p, but for only one is the orbit of a a packing.

The correct choice is simply described;

namely, it is

Here (12)a is obtained from a by dropping the line through 1 , and

is the unique permutation which interchanges a, p and fixes 1.

A

The other

cycles for which the orbit of a is a cyclic packing containing a, @ can be written down analogously.

We prefer, however, t o use the 8 permuta-

tions

Each of these is a cycle of length 7, and no two fix the same point of D. Also

'rri

maps a upon p or p upon a according a s i is even or odd.

Finally, the orbit

of

2

is a (non-projective) cyclic packing

under conjugation by A',

containing

a, p.

Next we define

ek = s o that go = a ,

el

an^

8

o f kS6,

= @ , and the Bk make up

%

=

2 o.

We a l s o define

R. H. Bruck

for a11 integers i and verify that the elements Ai, BI (0

5- i <_-

7)

and C0,C1,C2~C3 are the 20 distinct spreads disjoint from

a or f3.

a, f3 which do not commute with

In particular,

In addition we find that A1 =

e6,

B2 =

e5.

Consequently we have:

d O = qa,@ : ;AO,Al The other

2

; Bo,B2;

Coo

are obtained by successive addition of 1 t o the subscripts

of the A's, B1s and C1s, subject t o (*). Now we s e e that: a , B , A ~ lie only in

qO,2,;

a, B, B~ lie only in

0,

q6;

a, p, Co be only in

0,

q4-

And this i s enough t o prove that N(3) = 2. The rest of the proof can only be indicated.

Of the 19 spreads dis-

-R. H. Bruck

joint from a, p , distinct from Ao, which commute with neither of a, p, exactly 9 are disjoint from Ao, and one of these commutes with Ao. the remaining 8, four lie on the four on

2

a.

and four on

2 7.

Of

Moreover, each of

is disjoint from exactly one of the four on

ZT

The situation is similar for Bo (or Co), except that there are two spreads (instead of one) disjoint from a, @ and from Bo (or C ) which do not commute with a or p but commute with Bo (or Co).

Q

Consequently, we may conclude that

and that N(5) = 0 or 1. As a n instructive exarhple of a n anti-commutative s e t of 5 disjoint spreads, consider the following:

Since

2

2

contains a,p,AO,Al

and

ql

contains a,p,A l,B1,

and

contains a. @,Ao,B1, the 5-set is disjoint and anti-commutative.

However, among the 8 lines not covered by the spreads of the 5-set are the lines (46). (48), (68).

Since no two disjoint spreads can cover these

three lines, the 5-set is a maximal disjoint set.

In particular, then, we

have a n illustration of the c a s e N(5) = 0. This proves (xv)and completes our discussion of the complete

R. H. Bruck From (ii), (v), (x), (xi), we have

with equality only if there are no further types of packing of D.

IV. m = 4.

Here

There is indeed a projective plane of order n = 8, (a unique one) and it has a t least one complete ovaL We need only refer t o $2 for the case n = 2m = 8, e = 2, s = 4, t = 2.

What we want t o show here is that the

-

packings studied for the previous c a s e m = 3 are related in a significant way t o the problem of constructing a complement for a complete 10-point. Let D be a complete 10-point and let D* be a complement of D, s o that D,D* determine the unique projective plane n of order 8, and D appears a s a complete oval of n.

Let L be a line of .rr whfch meets

D and hence contains 'two distinct points of D. D.

That is, L is a line of

Let Do be the complete 8-point obtained from D by removing the

common points of L and D.

Then each point P of L which is not in

D determines a spread of Do, and the 7 spreads s o obtained constitute a packing

@

of Do.

Conversely, by adding the line L of D t o each

of the spreads of the packing which specify the points of i '

@,

we get a collection of spreads of D

which lie on L but are not in D.

The nature of the packing @

of the complete 8-point Do will

depend both on how D is imbedded a s a complete oval in n and on how

R. H. Bruck the line L has been Chosen. from $3.

Thus

where

=

Let us take, for the imbedding, the example

Since t = 2, the

(4) i s a "conicu and K i s its knot

group of a l l collineations of

T

which map D upon itself fixes K and

has a (normal) subgroup of index e = 3 which i s sharply triply-transitive

re

on the 9 elements of

First let the line L contain K and hence have exactly one point in common with

\& .

Then the subgroup, G(@), of S8 which maps

upon itself has a subgroup (of collineations of

T)

doubly transitive on Do and has order 8-7 3.

for some positive integer N. incidentally, N = 8.

Thus

which i s (at least)

Therefore

must be projective.

And,

Note that L i s acting a s the plane at infinity for

the affine 3-space determined by

p.

Next let L contain two distinct points of Nevertheless, G(Q)

contains K.

@

&

. In this case,

Do

has a subgroup (of collineations of n)

which fixes K, i s transitive on the remaining 7 points of Do, and has order 7 3.

Thus

for some positive integer N. N = 64 or (b)

@

Hence either (a)

@

i s projective and

i s non-projective cyclic and N = 2.

By a closer

examination, one may verify that the correct alternative i s (b). These remarks should suggest that there was some point in study-

R . H. Bruck ing the spreads and packings of the complete 8-point, even though we know in advance that it has no complement. And we can go further.

No one knows whether there exists a pro-

jective plane of order 10, even if we insist on one which has a complete oval (a complete 12-point). involve a careful study

4.

A first step in deciding this question should

of the packings of the complete 10-point.

(k = 3. ) Steiner triple systems.

1-design with k = 3.

A Steiner triple system i s a

Such a design D satisfies

If D i s to have a complement, Dl, then 3/v and hence

and

where (since k' > 1) m i s an integer greater than 1. Our basic question can be phrased a s follows: For what positive integers m does there exist a projective plane n of order n = 3m with a pointset D of order v = 3m t 2 imbedded in

T

such that D i s a

Steiner triple-system of order v in the sense that each line of n contains either no point of D or exactly 3 distinct points of D ?

R. H. Bruck For m = 2, n = 6, there is no plane whatsoever.

Nevertheless, it

might be of interest t o examine the spreads and packings of the various Steiner triple systems of order 15.

These systems of order 15 have been

listed several times (there are 80 distinct isomorphism classes) and could certainly be examined. For m = 3, the problem becomes more challenging.

On the one

hand, there are a t least 3 isomorphic planes of order 9 (the Desarguesian plane and two planes originally constructed by Veblen and Wedderburn, belonging t o the c l a s s e s of Hall planes and Hughes planes).

On the other

hand, the number of non-isomorphic Steiner triple systems of order 21 is very large (Doyen estimates that there are more than two million).

Hence

it is unclear how best t o attack the problem. One answer (not relevant t o m = 3) is t o limit attention t o nice c l a s s e s of triple systems.

For example, there are just two c l a s s e s of

Steiner triple systems for which the automorphism group is transitive on ordered triangles.

Consider t h e system of points and lines of a finite-

dimensional projective or affine space over GF(q). For the one class, we use projective space with q = 2.

For the other c l a s s , we use affine space

with q = 3. However, it seems more natural a t this point t o generalize the problem by not insisting on q = 2 or 3.

R. H. Bruck 5.

(k = q or q

+ 1. )

Affine and projective spaces.

Let q be

any prime-power, and let s, t be positive integers. By the affine design AD(s, q) we mean the system consisting of the points and lines (used a s varieties and blocks, respectively) of the s-dimensional affine space AG(s, q) over the finite field GF(q). By the projective design PD(s, q) we mean the design of points and lines of the s-dimensional projective space, PG(s, q), over GF(q). Both of these designs have

A = 1.

.The divisibility conditions k (v, necessary for the existence of a complementary design, is satisfied by AD(s,q) for a l l s but requires s t o be odd i n the c a s e of PD(s,q).

Thus we take s = 2t

+1

in the follow-

ing comparison:

+ 1, q): PD(2t + I , q):

For AD(2t For

For either:

- l)/(q k1= q(q2t -

- 1) .

k = q, kt = (q2t

+ 1, n = q(q + l)(q2t - l)/(q2 - 1) k=q

.

-

1)

.

Clearly if, for some positive integer t, one of AG(2t.-+ l , q ) , PD(2t

+ 1, q)

has a complement, then there exists a projective plane of

the order n given above.

For t > 1 this clearly poses two distinct

problems, since the parameters of PD(2t plement t o AD(2t + 1, q). (5. 1)

+ 1,q)

are not those of a com-

The c a s e of t = 1 is l e s s clear:

Parameters of PD(3. q): 2 2 2 v = ((2 + 1)(q + 11, b = (q + 1)(q k = q + l , r = q .2' % q + l , A = 1 .

+q+

1) ,

R . H. Bruck (5. 2)

Parameters of AD(3, q): 3 2 2 v l = q , b l = q (q t q t l ) ,

(5. 3)

Common order: n = q(q

+ 1) .

That is, each of AD(3, q), PD(3, q) has the parameters of a complement of the other.

Nevertheless, there is no reason t o assume that if one has a

complement, the latter i s isomorphic t o the other.

(I made such an assump-

tion in my 1963 Saskatoon lectures, but now regard it a s unwise. ) In the rest of the paper, we restrict attention t o PD(3,q), thus leaving a great deal untouched.

6.

Regular spreads of PG(3.q).

Constructions of packings.

In

Bruck [ 3 ] I exploited a connection between regular spreads of PG(3, q) and Miquelian inversive planes (represented in terms of t h e points and 2

projective sub-lines of order q in the projective line over GF(q ). ) But Miquelian planes have other representations, for example in terms of the points, lines and circles of a n affine plane over GF(q).

I now wish t o

exhibit a construction in which a regular spread of PG(3, q) is related i n a natural geometric way t o q2

-q

Miquelian inversive planes, two in the

I1projective line1' representation and the rest (if q > 2) ,in the "affine plane1! representation The construction t h e n can be used t o obtain a n alternative approach t o the packings of R. H. Denniston 1141.

I should

add that i t was Denniston's work, together with my desire t o avoid his use of the Klein quadric, which led me t o the present results. I shall repeat just enough from Bruck [ 3 1 t o make the present proofs understandable.

R. H. Bruck 2 We may imbed C = PG(3, q) i n C1 = PG(3, q ) s o that C consists of subsets of the points, lines and planes of C' under the incidence relation of C.

Then C 1 h a s precisely two types of points, those i n C and

those not i n C.

Similarly, C1 has precisely two types of planes.

With

/

a line L of C 1 is associated a non-negative integer n such that L contains exactly n points of C and l i e s in exactly n planes of C; and n can be q

+ 1,

1 or 0.

through each of the q The set,

2

If a line L of C1 contains no point of C then

+1

points of L there passes a unique line of C.

of these lines of C is a spread of C (that is, of

J(L),

PD(3,d) and is regular in the following sense: For each line M of C

d(L),the set, & , consisting of the q + 1 lines of d(L)which meet M, is a regulusof C. That is, & i s a s e t of q + 1

which is not i n

disjoint lines of lines of

bL

C such that every line of C which meets 3 disjoint

&.

meets every line of

that, for every regular spread ef

of. C, there are precisely two distinct

lines L of C1 of type n = 0 such that Theorem 6. 1. A be a line of C.

one of the q2 (i)

lines of

.TI

-q

Let

We shall have t o repeat the proof

$=

J(L).

be a regular spread of C = PG(3,q).

Let

Let C be imbedded i n C1 = ~ ~ ( 3 . qand ~ )l e t n be planes of C1 which contains A but i s not i n C.

Then:

can be chosen in 2 distinct ways s o that the q2 distinct

d,distinct from

A, meet n i n q2 distinct points of a line

L of n of type n = 0. The remaining point, SL , of L is on A (but not in C. ) The points S2.0 ', obtained in t h i s way corresponding t o the two choices of

w , are distinct and may be -called the circular points a t

infinity corresponding t o

d

and A.

R. H. Bruck (ii) If q > 2,

71

can be chosen in

ways s o that the q 2 distinct lines of

d,distinct from

the q2 distinct points of a n affine (Baer) subplane, q, with A a s the line a t infinity of p.

The q

+1

A, meet n i n .

'p, of

R, of order

points of A which

a c t a s the points a t infinity for p are not in C and are distinct from the circular points

n8SlI.

Each regulus of

$

which contains A determines

(and is determined by ) q distinct points of p lying on a line of p. regulus of

$

Each

which does not contain A determines (and i s determined

by) a circle of p; that is, a conic of

n which meets A in the circular

points

+1

n , n and contains exactly q Proof. -

We represent

distinct points of p.

C = PG(3, q) in terms of a 4-dimensional

vector space V over GF(q); points, lines and planes being represented by I-dimensional, 2-dimensional and 3-dimensional subspaces, respectively, of V w e r GF(q).

Once we have chosen a basis

of V over GP(q), we shall be especially interested in the line

and the lines

one for every 2x 2 matrix X over GF(q). The line L(X) i s skew t o L(-), and every line skew t o L(-)

has form L(X) for a unique X

Two

R. H. Bruck lines L(X),L(Y) are skew if and only if the matrix X

-Y

is nonsingular.

We may assume without l o s s of generality that the b a s i s

(6.1)

has been chosen s o that

where I denotes the

2 X 2 identity matrix and U

a (fixed) irreducible

2 X 2 matrix over GF(q). ~ e ht be a characteristic root of U.

Then A is i n G F ( ~ ' ) but

not in GF(q), and h q is a second characteristic root, distinct from X. Clearly u12

#

0, e l s e U would be reducible.

Hence the equations

determine a pair of distinct conjugate elements,

0 ,crq,

2 of GF(q ).

We

note that (6.6)

L

U12U

- (ull - uz2)u -

UZ1

= 0

.

and similarly with u replaced by uq. Next we imbed C = PG(3, q) in Ct = PG(3, q 2) by extending V t o

a vector space Vt over GF(~') having the same basis (6. 1) a s V.

In

addition (to complete the imbedding) we identify a subspace of Vt over G F ( ~ which ~ ) happens t o have a b a s i s of elements of V with the corresponding subspace of V over GF(q). For each element, p , of GF(~') we define the plane

C' by

n ( p ) of

R. H. Bruck A necessary and sufficient condition that

that p be not in GF(q).

~ ( p )should not be i n C is

Henceforth we assume:

Then

and the only ~ o i n t of s C contained in n(p) are those points of C contained i n A. 2 To keep things clear, let M2(q), M2(q ) represent the s e t of a l l 2 x 2 matrices with elements in GF(q), GF(~') respectively. For each X in M2(q), the line LO[) of C meets ~ ( p )in the point

where v(p;X) is the vector i n Vl defined by (6. 11)

v(p;X) = (px,,

+ xzl)el + (px12 + ~

~

~

)

In particular, the "point a t infinityf1

is a point of A for each X

Specifically (for X

c

P

M2(q), X

#

0, and may or may not be i n C.

M2(q),X # 0) P(.a;p,X) is 'in 7, precisely when X i s

a singular matrix. If X,Y are in M2(q), X # Y, the vector subspace of Vf over 2 GF(q ) containing the points P(p;X), P(p;Y) i s the same a s the vector subspace containing P(p;X) and the point (at infinity), P(m, p;Y

- X).

e

~

R. H. Bruck Thus P(p;X), P(p;Y) are distinct and the line

Lm

has type n = 0 precisely when the lines L M . Now we are ready t o study the q 2 points

of

C are skew.

These are g2 distinct points of n ( p ) . They will lie on a single line of 7~ ( p )

precisely when the vectors

2 are linearly dependent over GF(q ). Since o and a q satisfy ( 6 . 6 ) , the linearly dependent case corresponds t o p = u or u'I The relevant points

.

a t infinity corresponding t o = ( a ) , =(aq) are

These are points of A which are distinct and not in C

. This proves (i).

For the proof of (ii) we strengthen (6. 8) t o

For (6. 17) t o yield anything, we require q > 2. the affine subplane lines joining them.

@ ( p ) consisting of the

First we must establish

g 2 points (6. 14) and the

It i s easy to s e e that a line of

.rr(p) joining two

distinct points (6. 14) contains exactly q distinct points (6. 14). More-

R. H. Bruck

over, the set of a l l points of A lying on such lines consists of the q

+1

distinct points

P(-, p;I) and P(-, p;aI

(6. 18)

+ U),

a c GF(q)

.

None of t h e s e is i n C, and (especially i n view of (6.23) below) each is distinct from ~2 and

S2

I.

This is enough t o show that p (q) is an affine

subplane of order q with A a s its line a t infinity. Next we appeal t o the known fact that, if X1 ,XZ are elements of M2(q) and X2

- X1

C containing

is non-singular, the unique regulus of

the three skew lines

consists of A and the lines LO[) with X of form

~ ~ p l this ~ ifact n t~o the lines other than A of

$

regulus of t h e lines of

E!,

we s e e that the

which contain A correspond in a one-to-one manner t o

p (q).

Finally, consider any matrix K i n M2(q) such that the line L(K) is not in

.$.

Equivalently, the matrix

is nonzero for a l l choices of a, b in GF(q). Since L(K) must meet exactly q regular, t h e s e q

+1

+1

distinct lines of

d

8.

is a spread,

Since

lines must constitute a regulus,

Since L(K) is skew t o A = L(m),

@.

does not contain A.

is

, of

d

Hence

.

&

R. H. Bruck consists of q

+1

distinct lines of form

L(a1 + bU), a, b

c

GF(q), such

that

Conversely, t o every regulus

of

L!

which does not contain A,

there corresponds a t l e a s t one matrix K in M2(q) such that L(K) is not in

$

and such that

d?

consists of the lines L(a1

+ bU),

a, b

c

subject t o the (non-homogeneous) determinantal equation (6.19). follows a l s o that

8

GF(q)

It

corresponds to the s e t of a l l points

of ~ ( p )subject t o (6. 19). The equation (6. 19) a l s o determines a point-set of ~ ( p )consisting of a l l points

subject t o the homogeneous determinantal equation

2 Here x, y, z are elements of GF (q ), not a l l zero. t h e points (6.21) with x, y in GF(q) and z = 1. course, the equation of a conic in

The points (6. 20) are Equation (6. 22) is, of

~ ( p ) . The conic meets the line A in

the Itpoints a t infinity" obtained by taking z = 0 , y = -1 i n (6.22). yields

whence either

This

R. H. Bruck

or x = hq.

Hence the "points a t infinityu of the conic are

Since 1,lqare not i n GF(q), neither point is a ljoint at infinity for p.

-

hql

-

In (6.22), we have, for the first time, used matrices, hI U and 2 U, which are i n M2(q ) but not i n M2(q). We shall show in a

moment that

As soon a s (6.23) is established, the proof of (ii) and Theorem 6.1 will b e complete.

Indeed, l e t Pl,P2, P3 be three distinct points of the

affine subplane p (p).

The corresponding lines L1, L2, L3 of

mutually skew and hence determine a unique regulus,

&? , of

d

are

d. fl

,P ,P d o not lie on a line of p (p). Then 'L 1 2 3 Hence & determines a unique conic of .rr which

Assume, further, that P does not contain A. contains q

+1

distinct points of p (p), including P1, P2, P3, and meets

A i n the points (6.22). 52,Q

If (6.23) holds, the conic contains P1, P2, P3,

and is determined by t h e s e 5 points. To prove (6.23), we note that

where, since h = ou

12

+

u22*

R. H. Bruck

and

Hence

and

Similarly for the second formula of (6.23). This completes the proof of Theorem 6. 1. The lemma which follows is a corollary of Theorem 6. 1. Lemma 6.2.

Let

A

and let A be a line of

.

be a regular spread of

I: = PG(3, q), q > 2,

Set

and let

@-,,i 5- i r-. n , be the n distinct reguli of

$1

which contain A.

Then there exists a

R. H. Bruck

set of n regular spreads

of

C with the following properties: (a) For each is the only lines common t o

J

and

4

are the

Ri.

lines of

Y s con-

(b) Each line of C which is skew t o A but not in tained in exactly one of the

fi.

Proof. We assume the situation of Theorem 6. 1 and take

.rr t o

d

be a plane of C 1 through A which is met (see (iii)) by the lines of

distinct from A in the points of a n affine subplane, p , of .rr with A a s its line a t infinity.

The corresponding projective subplane p

*. a s a

sub-

plane of order q in the projective plane n of order q2, is a Baer subplane. line of

That is, every point of .rr which is not in p

*

* p .

Each of the n = q 2

+q

l i e s on a unique

lines of p , considered in terms of the

points of p alone, determines a unique regulus of

$

which contains

A and, considered a s a line of n which contains no point of

the regulus in a regular spread of C.

Thus we get the n reguli

each imbedded in a unique regular spread

and

Yi

di and hence

This proves (a). If L is not in

not be A.

* p .

(Ri.

Therefore L is in

@i.

then L meets n in a point P

Hence P l i e s on a unique line of p

Therefore I l i e s in

proves Lemma 6.2.

If L is common

then L meets n in a point of p lying on the line of

@ which determines

which is not in

&is

Ji.

Now consider a line L of C which i s skew t o A. to

C , imbeds

<

*, and this can-

for a unique choice of i.

This

R. H. Bruck As Denniston [14]remarks, the 240 ptickings of PG(3,2) (q = 2)' are well-known.

They will not be considered here.

The theorem which

follows, although it asserts.more than Denniston [14], was stated in present form by Denniston a t the 1972 conference in Bressanone. (Denniston) If q > 2, C = PG(3, q) has at l e a s t two 2 inequivalent types of packing formed from one regular spread and q + q Theorem 6. 3.

spreads which are subregular of index 1. e f .

Consider the regular spreads

For each i = 1,2,.

6?

..

,n, let

xf

and

di of Lemma 6.2.

; be obtained from di by replacing '

&b

by the opposite regulus,

d dl

Thus

is a subregular spread

of index 1 (for terminology, s e e Bruck[ 3 1. ) We now demonstrate that the collection

of 1 + t = q

2

+q+ 1

spreads is a packing.

and hence is in nod;

but meets every line of every a line L of

C distinct from A.

of a unique regulus and

A; for a unique

8,Yi

is in

B;, the line

i.

Next consider

d

containing A, and hence is in

d?;

If L is skew t o A then there are two cases.

L is a l s o in

Ri

8;for some

d?;.

and thus is in no

Therefore L is in a unique

.

Since

But then L meets A, a

Thus (for L skew t o A) if L is in

$

i.

in common, and since no line of

L must be in

Finally, let L be n d i n

meets no

$

If L meets A then L meeis the lines

have only the lines of

contradiction.

d;.

i.

of

8,suppose that

If L is in

@i

Ri

First, the line A is in

d

then L is in no

Then L is skew t o A (whence L &;)

B>

and L is in

di for a unique.

R. H.. Bruck

@

Now we have provedthat c a s e i = 1) that Let

>(and

dland @i is in

is in

of replacing a packing.

t=q

d !

J1)and @i is in 8;.

be the spread obtained from

J1

2

is i n

is a packing of 2. We note (the

+q

$, 8;in @ Moreover,

@

by

and

by reversing Hence, if

$I.

dl,

P*

a)*

(PI. Thus is the result

dlrespectively, @*

is a l s o

each comprise 1 regular spread and

subregular spreads of index 1. However, the regular spreads,

8, 8,are differently related t o the subregular spreads

since the reguli

.

Q2,.. , Rt

of

d are opposite t o the reguli

whereas, for each such I, no regulus of regulus of

8;. Therefore

@, (P*

J1can be opposite t o a

are inequivalent.

Next I wish t o treat briefly a different type of collection of spreads. Recall that if PD(3, q) has a complement (in the s e n s e of $1) then, in particular, there exists a t l e a s t one collection,

cs,of spreads of

C = PG(3, q) such that every pair of skew lines of C l i e s in exactly one member of

cs. I t occurred t o me t o wonder whether such a

be the orbit of a spread

$

under a suitable group,

$,

could

a,of collineations

of C. This brought me, in time, t o the concept of an admissible pair

R. H. Bruck (

a), subject to the following conditions: (i) $ is a spread and a is a group of collineations of

$,

Z = PG(3, q).

(ii)

d

i s transitive on the set of all ordered pairs of skew lines

of C. (iii) If 8 is i n

8 and if the spreads 28,

than one common line, then

$=

Given such a pair ( J

J 8 have more

r88.

,a), or given a pair subject t o (i), (iii)

and to (ii) weakened by dropping the word llorderedll, then the orbit of

$

has the above-mentioned property that every pair of skew

under

lines of C lies in exactly one member of the orbit. Hearing of the concept of a n admissible pair ( $,

) by word

of mouth, various geometers expressed deep skepticism a s to existence but no one offered a proof of non-existence (at least, so far a s I know). Finally, in 1970, 1 turned the problem over to my Ph. D. student Sung-chi Lin, who will receive his Ph,D. degree on this topic i n the spring of 1973. Some of his results may be summarized a s follows: An admissible pair

($.a) does not exist in any of the following

cases: (a)

i s regular.

(b)

is subregular of index less than (q - 1)/2.

(c) q i s an odd poMRr of 2 and the subgroup of upon

a mapping

d

i s the corresponding Suzuki group. There are other results but a complete proof of non-existence has

not been obtained. admissible pair.

On the other hand, I know of no example of an

R. H. Bruck

Another topic which seems t o be relevant here concerns a method of constructing an affine plane a of order q2 i n terms of certain spreads of C = PG(3,q). We choose a line A of C and, for the point-set, take the s e t of a l l lines of C which are skew t o A. will be partitioned into two disjoint sets,

(p , of

a , we

The line-set of a

and

, whme

t h e set of a l l planes of C which do not contain A and where

&

is is a

collection of spreads of C containing A, with properties t o be discussed. The containing relation will serve t o decide whether a line L in

@

( a s a point of the proposed affine plane a ) is incident with a plane i n or spread in A plane

( a s a line of a). in

T

3

meets A in a unique point Q and hence con-

t a i n s precisely q2 distinct lines in A spread lines i n

in

y , every two of which intersect.

contains A together with precisely q2 distinct

2,every two of which are skew.

.

Hence a , if it exists,

must have order q 2

Let L, M be distinct line of C, both skew t o A.

If L, M intersect,

they lie i n no spread of C and they lie i n a unique plane, LM, of C. Moreover, LM does not contain A. a containing L and M. C.

If L, M are skew then they l i e i n no plane of

Hence, if a is t o be a plane,

taining L and M.

(I)

Thus LM must be the unique line of

must contain a unique spread con-

Thus, a t the very least,

e

must satisfy:

is a collection of spreads of C = PG(3, q) each contain-

ing a fixed line A ~f C.

.

(11) To every pair of lines L, M of C such that A.L. M are

mutually skew, there corresponds one and only one spread i n

which

R. H. Bruck

contains L, M. Now we wish t o show that (if

satisfies (I), (11)) then a ,

with points, lines and incidence a s defined above, i s indeed an affine 2 plane (of order q ). We know that every two distinct points of a lie on a unique line of a. Next we wish t o discuss the parallel classes of lines of a.

Let .rr be in

3 , s o that

.rr i s a plane of C meeting A in a

unique point, say Q, of C. If .rrl i s a plane of C not containing Q (and hence not containing A) then the line of intersection of .rr, i s skew t o A; hence .rr ,.rr spread in

are intersecting lines of a.

L i s distinct from A.

now see that titioned into

of the q'

Since .rr does not contain A, then

i s a spread containing A, L, with A

#

L,

are intersecting lines of a. We

Hence n ,

3 (the s e t of a l l planes of C not containing A) is parq + 1 parallel classes of lines of a. Indeed, for each of

distinct points Q of A, there is a parallel class consisting planes of

C which contain Q but not A. We also'see

we hope t o prove about (111) each.

$

Since

then L i s skew t o A.

+1

is a

then (by a general property of spreads of finite 3-space)

contains a unique line L of .rr.

the q

If

in C

2

what

, namely:

i s partitioned into q2

Two distinct spreads i n

2

-q

disjoint classes of q2 spreads

belong t o the same c l a s s if and only if

they have only the line A in common To show that (I), (II) imply (111) we need some counting arguments. Let Q be a point of A.

Ifs% i s a line of C skew t o A there i s a unique

plane n which contains L and Q and hence does not contain A. If n i s a plane containing Q but not A, then n contains exactly q2 lines skew t o A; and there are q2 such planes .rr.

Hence the above-mentioned

R. H. Bruck

behave a s they should.

parallel c l a s s e s of lines in the number of distinct lines of C

In addition,

skew t o A is

Next, if L is a line of C skew t o A, we need the number of lines M of

C skew t o both of A,L. containing Q and M.

Given such an M, l e t .rr be the plane of C Then .rr contains neither A nor L.

Hence .rr

meets A in Q and L in a point R, and therefore. contains precisely

lines skew to A and L Letting .rr range over the q2

- 1 planes of

C

which contain Q but neither A nor L we see that the number of lines of

C skew t o A and L is

Now, by using these results, we see that the total number of spreads in

c

is

and that, for every line L skew t o A, the number of members of

con-

taining L is

Finally, let

$

be a spread i n

skew t o A and not in and only one spread

and let L be a.line of

C which is

.

We prove (In) by showing that there is one

in

which contains L and has only A in

$

R. H. Bruck

3 . First we obsenre that

common with

L meets

tinct lines of C, necessarily distinct +om A. with L in a spread.

lines of

d

$

in q

+1

dis-

None of these can occur

There remain precisely

which are skew t o L and A.

Each of these is contained

, and no two in the

with L in a spread in

same spread. This

accounts for

distinct spreads in

which contain L There remains in

one additional spread

$

i n common with

$

.

which contains L, and

exactly has only A

To sum up: /I). (II) imply QIIL.

In the presence of (I), (II), (III), it should be obvious that a is 2 an affine plane of order q

.

In my 1963 Saskatoon lectures (Bruck [ 2

1) I touched on the

above

(potential) constructjon of an affine plane a of order q2 and gave one example, in which a was Desarguesian.

Later, i n the fall of 1963,

T. G. Ostrom sent me (in a private ,letter) ample evidence that "every known plane of order q2" had such a representation

So far a s I know,

0 s t ~ 3 - 1results 1 ~ ~ have never reached print in this form.

I cannot presume

t o publish them here but what I will do is connect the topic with a realm of ideas very close t o Ostromfs present work. As in the situation of Theorem 6. 1, we imbed C = PG(3, q) in Cf = PG(3,q 2) and choose a plane n of Ct which contains the given line A but is not in C.

obtained from

TI

Let no be the Desarguesian affine plane of order q2 by deleting the line A and i t s points.

For every pair

R. H. Bruck P,Q of distinct points of n o , let PQ be the line of n o through P and' Q, considered a s a s e t of qZ distinct points of no.

By t h e point a t

infinity on PQ we shall mean t h e point of intersection of the corresponding line of .rr with

R

There is a natural correspondence between the set of points of n and t h e set

@

of lines

@

on a unique member of point of no.

0

C skew t o II Indeed, e a c h point of n o lies and e a c h member of

@

meets n in a unique

We shall u s e t h i s correspondence.

As Ostrom h a s frequently remarked, any affine plane of order q

2

can be regarded a s having t h e same point-set a s t h e Desarguesian plane

roe Consider any affine plane n b on the same point-set a s n o and, for every two distinct points P,Q of *r0,let (PQ)' b e t h e line of n b through P,Q, considered a s a set of q2 distinct points of no.

The

following condition turns out t o be necessary and sufficient that the above natural correspondence be a n isomorphism of .rrb upon one of the planes

a defined in terms of a collection (#) If P,Q

, subject t o

(I), (11):

are distinct points of r o such that the point a t

infinity on PQ i s in C, then

(PQ)' = PQ.

This condition i s perhaps deceptively simple, in that we are assuming that

rb

is a plane of order q

points of n o and if (RS)'

. Thus, if

2

R, S are two distinct

RS then, for every two distinct points P,Q

of (RS)', the point of infinity on PQ i s not in C.

For, if, for some such

P,Q, t h e point a t infinity of PQ were in C, we would have R,S and' hence

E

(RS)' = (PQ)' = PQ

R. H. Bruck

in contradiction t o the necessary assumption that the point of infinity on RS is not in C. Now let us examine (#).

First let P,Q be two distinct points of

0 such that the point of infinity on PQ, s a y the point R of A, is in C. Then the line PR of T contains exactly one point, namely R, of C and .rr

hence l i e s in exactly one plane of C.

The latter is in

scription of a). Conversely, each member of

2

2 (see our de-

meets .rr in a line

containing exactly one point of C, namely a point of A.

Thus the common

line (PQ)' = PQ of n o and .rrb is represented by a unique member of

3

. Next assume that the point, R, of infinity on W

i s not in R.

Consider the line (FQ)' of .rrb, which may or may not be equal t o FQ. For every two distinct points S,T of (PQ)', the point of infinity on ST is not in C.

This means that the line of .rr (not just n o ) containing

S, T has no points of C.

In particular, the lines of C through S, T are

skew t o each other and t o A,

Hence the q2 lines of C corresponding t o

the points of (PC))', together with A, form a spread

A(P,Q).

It should now be clear that, given .rrb subject t o (#), we can construct

e

subject t o (I), (11) s o that a is isomorphic t o r i,.

versely, given a and hence given

Con-

subject t o (I), (11), we can con-

struct n b isomorphic to a. For the Desarguesian case, we simply take (PQ)' = PQ for every pair of distinct points P,Q of nb.

In general the point-sets

(PQ)' may

be more complicated; some,, for example, may be affine subplanes of n o of order q (with the line A a s line a t infinity and with no points a t infinity in C).

R. H. Bruck

One l a s t remark: Instead of attempting t o construct n b . or a l l the spreads i n

e , we might look for new ways of constructing a single

spread of C i n terms of the circle of ideas surrounding condition (#).

7.

Translation planes of order qL. In t h i s section I wish t o

indicate (without proof) some improvements i n that theory of classification of translation planes which was developed in Bruck .[3]. To begin with, we need t o know that there e x i s t s a construction process which assigns t o e a c h spread $ of PG(3,q) a translation plane n($ ) of order q

. Next we should know that, when

2

q is a

prime, every translation plane of order q2 h a s such a representation ~ ( x f ) . On the other hand, when q is not a prime, a translation plane n

has a representation =($)

precisely when .rr is, i n a precise sense,

d-dimensional over GF(q) where d = 1 or 2.

(d = 1 only for the

Desarguesian plane. ) I have i n mind the construction process developed by Bose.

(See

Bruck and Bose [7]. ) However, Segre [12] developed a n equivalent process a t about the same time and

- t o our intense

surprise

- we found that we

had been anticipated by ~ n d r ; [I], i n slightly different language, but in a paper which must have been read by a l l of us. Whatever the construction process, one would like t o be assured that one can characterize the translation planes n($) i n terms of the spreads

d.

Thus the following theorem is crucial:

Theorem 7. 1.

(~ffneburg.) Let

d,$

be spreads of PG(3, q).

A necessary and sufficient condition that the translation planes n ( J ) ,

n ( J f ) be isomorphic is that the spreads

$,

be equivalent; that is,

R. H. Bruck

that there exist a collineation of PG(3, q) which maps

bB

upon

I.

Theorem 7. 1 calls for a classification of spreads of PG(3, q) into equivalence classes.

Such a classification h a s been completed only for

q = 2,3.

For q = 2 there is one equivalence c l a s s (all spreads are

regular).

For q = 3 there are two classes; one consists of the regular

spreads and the other of the subregular spreads of index 1. For q > 3 many examples are known but the only extensive classification is that of the subregular spreads, which I shall now discuss.

8 , of

Recall that a regulus,

PG(3,q), is a set of q

+1

(dis-

tinct and mutually) skew lines of PG(3, q) with the property that every line of PG(3,q) which meets 3 distinct lines of them, The set, a regulus I

I,

consisting of the q

d? , is a l s o a regulus, and

cover t h e same points.

regulus,

&'

&

% I

, and

if

versely,

d

$

Hence, if

%

by

@

I)

has been obtained from

!L.

can be obtained from

Let u s say that two spreads provided that

d 1can be obtained

sequence of simple reversals. lence relation

=

(&')I

distinct transversals t o

@.

Moreover,

and

is a spread containing a

is the line-set obtained from x ! by reversing

$I

(that is, by replacing

say that

+1

meets a l l of

then

dt

i s a l s o a spread.

by a simple reversal.

We

Con-

by a simple reversal.

$, from

$I

d

are reversal-equivalent by a finite (possibly empty)

Then reversal-equivalence is an equiva-

True, it is only of interest in connection with spreads

which contain a t l e a s t one regulus and (answering a conjecture in Bruck and Bose [7] in the negative) there are now many examples of spreads which contain no reguli.

Equally true, other ways of generating spreads,

apart from using simple reversals, are known.

Nevertheless, reversal-

equivalence is very useful and provides many interesting unsolved

R. H. Bruck problems. Recall, next, that if A, B, C are three (distinct and mutually) skew lines of PG(3,q), there e x i s t s one and only one regulus of PG(3, q) which contains A, B, C.

A spread,

$ , of

&(A, B, C)

PG(3, q) -if,

called regular provided that, for every three distinct (and hence skew) lines A, B, C of

$, every line of the regulus

As is well-known, the spread plane .rr ($)

$

$.

@(A, B, C). is in

is regular if and only if t h e tsanslation

i s Desarguesian.

A spread,

$, of

PG(3, q) is said t o be subregular if it is ,

reversal-equivalent t o a t l e a s t one regular spread, and t o have m

x k

(where k is a non-negative integer) if it can be obtained from some regular spread by a sequence of k simple reversals but cannot be obtained from any regular spread by a shorter sequence of simple reversals.

As a

consequence, if, for some integer k _> - 0, there e x i s t s a subregular spread

zd of index

k, then every spread

$I

equivalent t o

$

the collineation group of PG(3, q)) a l s o is subregular of index k.

(under The

subregular spreads of index 0 are the regular spreads and form a single equivalence class.

There are no other spreads for q = 2.

For q

>

2,

subregular spreads of index 1 exist and form a single equivalence class, corresponding t o the Hall planes.

For q = 3 there are no other spreads,

and for q = 4 there are no other subregular spreads.

However, for q _> - 5,

subregular spreads of index 2 exist but do not form a single equivalence class. and k

(In Bruck [3], tables are given which show the facts for q 5- 11

<-

5, along with additional information )

A serious effort a t classifying subregular spreads rest upon t h e fact that a regular spread wf of PG(3, q) may be regarded a s a ~ i q u e l i a ninversive plane M(q).

It is easy t o show the following about

R. H. Bruck

$:

a regular spread

$

(i)

distinct reguli.

contains exactly q

2

+1

Each regulus consists of q

+ 1)

distinct lines and q(q

+1

Three distinct lines A, B, C of

(ii)

$

distinct lines of

.

$ lie in one and only one

J.

regulus of (iii)

Let

@

such that A is in only one regulus common with

d?

.

If

dR

(iv)

be a regulus of

$, and

but D i s not in

@.

is a regulus of of the lines of

A, D be lines of

$

Then there exists one and

d which contains

of

permutation 9 = 8(@) reguli of

2

D and has only A in

J, there exists one and only one $

$

which maps reguli of

upon

d , fixes every line of d?, but is not the identity permutation.

Moreover, 9 has order 2 and permutes the q2

-q

-

lines of

in

cycles of length 2. We call 9 = e(&)

inversion with respect t o the regulus

we say that two distinct lines A, B of

@

&.

And

are conjugate withrespect t o

provided that 9 = 9(&) interchanges A, B.

Two further prqperties are

of frequent use:

R2 are disjoint reguli of d , then Ql , % have one and only one common pair A, B of conjugate lines. If Ol, @ (v)

If

QI,

are distinct but not disjoint reguli of

$, then

ql, Q2

have no com-

mon pair of conjugate lines. (vi)

-1 q- 1

q

If A, B are distinct lines of

distinct reguli of $,,having

d,there exist exactly

A, B a s a conjugate pair.

reguli are disjoint, and each of the q2

-1

lines of

$

These distinct

from A, B lies in exactly one of these reguli. One fact seems worth recording in connection with (iv). Let

d?

R. H. Bruck

be a regulus of the regular spread of PG(3,q) with

(R,

d , and let 2

be the unique quadric

a s its two sets of rulings.

Finally, let $J

2 upon its upon the lines of d.

be the unique polarity of PG(3,q) which maps each point of tangent plane.

Then ( induces 9 = B(@)

Now we need the concepts of linear and non-linear sets (and of complete sets) of disjoint reguli of a regular spread.

Let

be a set of k 2 - I disjoint reguli of a regular spread

d. We say that

(7. 1) i s linear if and only if there exists a pair of distinct lines A, B of

Qi.

xf which are conjugate with respect t o each of the reguli (iv), (v), (7. 1) i s certainly linear i f k = 1 or 2.

However (for q _> - 5)

there are non-linear sets of three or more disjoint reguli. (7. 1) is complete i f k = q

- 1; in this case,

tinct lines A, B which are not in any of the

By

We say that

there exist exactly two dis-

gi. We may call

carriers of the complete set of disjoint circles.

A, B the

By (vi), any two distinct

can s6we a s the carriers of a (unique) complete linear set

lines of

of distinct reguli of

J.

(We leave aside, for the moment, the question

a s to existence of complete non-linear sets of disjoint reguli ) The following theorem is proved in Bruck [3]: Theorem 7.2.

Let

$

be a regular spread of PG(3,q), let

a complete linear set of disjoint reguli of obtained from

Define

$ by reversing

$, and let

some t of the reguli in

d

I

be

be a spread

&? , where

R. H. Bruck

k = Min(t,q

(7.3) Then

Ld

is subregular of index k.

Note that, in particular, if ing all of the reguli in the set, circles of reguli of

- 1 - t).

, then

c*, of reversed reguli, J*. And and from

d* i s obtained from xf by reversJ* is regular. It i s also true that is a complete linear set of disjoint

J1

is obtained both from

>8 *

by reversing q

- 1-t

by reversing t reguli of

*.

When q i s odd and

this dual origin of

causes some trouble (which can, however be

X !

overcome). The spreads,

dl,

which arise a s in Theorem 7.2 correspond to

the so-called ~ n d r dplanes (including the Hall planes for k = 1). The ~ n d r eplanes ' are completely classified in Bruck [3].

The classification

of these planes (and the more general ~lsubregularll planes) rests heavily on the following: Theorem 7. 3.

Let

,,/

I

be a spread of PG(3, q).

Let k be an

integer satisfying

Then: (i)

regular spread

If

d

58' is subregular of index

k, there exists a unique

of PG(3,q) and a unique set (7. 1) of k disjoint

reguli of PG(3, q) such that

d t i s obtained from d

by reversing all

R. H. Bruck of the reguli in (7. 1). Conversely, (ii)

If

i s 'obtained from a regular spread $ by reversing

$

a set (7. 1) of k disjoint reguli of

$ , then '!x

is subregular of

index k. Clearly Theorem 7. 3, in so far a s it applies, reduces the study of subregular spreads of index k t o a study of a fixed regular ~ p r e a d and the determination of the equivalence classes of s e t s of k disjoint reguli of

$ upon

$

under the group of all collineations of PG(3, q) .which map

d. But,

other problems.

for index k not l e s s than (q

- 1)/2,

there are

These have been resolved by my student William Orr.

In his Ph. D. thesis (Madison, Wisconsin, May 1973) Orr proves the following: Theorem 7.4.

(Om.) Let

be a regular spread of PG(3, q), and

let (7. 1) be a non-linear s e t of k disjoint reguli of 3 5- k 5 -q

- 1. )

Let

$I

be the spread obtained from

each of the reguli in (7. 1). Then the only reguli in reversed reguli

(Ri

@ 1 and

J.

(b) the reguli of

(Hence by reversing

d 1are

(a) the k

disjoint from t h e reguli

?f (7.1,. I had proved (Bruck [3]) an earlier version of Theorem 7. 4 in which

the hypothesis of non-linearity was dropped but k was required t o satisfy the inequality

This result was needed for Theorem 7. 3. we can now state:

Using Theorems 7. 2, 7. 3, 7.4,

R. H. Bruck Theorem 7. 5.

Assume q > 3.

of PG(3, q) of positive index. (I)

Let

Then one of the following holds:

$, and

There exists a unique regular spread

set (7. 1) of k > 0 disjoint reguli of

d

non-linear or (b) (7. 1) is linear and 2k

dl

be a subregular spread

$I

i s obtained from

d

a unique

such that either (a) (7. 1) is


- 1, with the property that

by reversing the reguli in (7.1).

In this case,

has index k. (11)

q is odd and there exists a unique pair

tinct regular spreads of PG(3,q) such that by reversing a complete linear set in a complete linear set

disjoint reguli of

is obtained from

of disjoint reguli of

of disjoint reguli of

of dis-

>B

$ (resulting

$*) and

$I

can

- 1)/2 disor from * by reversing a unime s e t of (q - 1)/2 *. In this case, $ has index (q - 1)/2.

be obtained either from joint reguli of

*

$*

-$, *

by reversing a unique set of (q

As previously remarked, for q = 2 every spread is regular. q = 3, the subregular spread of index (q

For

- 1)/2 = 1 is truly exceptional,

since it can be obtained from 10 distinct regular spreads by rwgrsing a regulus.

These remarks explain why we require q > 3 in Theorem 7. 5. If we ignore the case (11) of Theorem 7. 5, which has been studied

completely in Bruck [3], Theorem 7. 5 t e l l s u s that the problem of classifying subregular spreads (and the corresponding translations planes) reduces t o the classification of s e t s of disjoint circles of a finite Miquelian inversive plane M(q). But t h i s fact does n'ot end the problem; indeed, there is much t o be done.

Let us give a crude summary of some of the work of Bruck [3] in the

c a s e that

R. H. Bruch

e q=p i s a large prime-power, where p is a prime and e _> - 1 a positive integer. Consider only the subregular spreads of index 3, corresponding to a triple of disjoint reguli.

The number of equivalence classes corresponding to a

linear triple is asymptotic t o

and the number of equivalence classes corresponding t o a non-linear triple i s asymptotic to

Conclusion: For q large, there are enormously many inequivalent subregular spreads of PG(3, q) and non-isomorphic translation planes of

.

order q2

A s a first unsolved problem, there is the classification of non-

linear sets of 4 disjoint reguli,

Another natural topic is the classification

of maximal non-linear sets of disjoint reguli. The best result on the latter problem i s brand new: Theorem 7.6.

(Thas, Orr. ) Every complete set of reguli of a

regular spread of PG(3, q) is linear. A t the time of my Bressanone lectures, Thas announced that he had

proved Theorem 7.6 for q e v e n More generally, he had proved the case q even of the following theorem: Theorem 7 . 6

*.

(Thas, Orr. ) Every flock of an ovoidal inversive

plane w e r a finite field GF(q) i s linear.

R. H. Bruck

For q even, Theorem 7.6

* is more general than Theorem 7.6,

since Miquelian inversive planes are ovoidal but not every ovoidal inversive plane i s Miquelian. On the other hand, for q odd, Theorem 7.6

*

i s equivalent t o Theorem 7.6, since the ovoidal inversive planes coincide with the Miquelian inversive planes.

Thasls proof for q even does not

work for q odd, and Om's proof for q odd (which will appear in his thesis) does not work for q even. I hope that Thasls paper will appear in this volume just after the

1

present paper. I will close with a result of Orr for the case q = 9 which appears quite exceptionaL A complete (hence linear) set of disjoint circles of M(q) = M(9) would consist of 8 circles.

Orr discovered a non-linear

set of 7 disjoint circles, one of which is orthogonal to the other 6.

The

collineation group mapping the 7 circles upon themselves fixes the orthogonal circle and is doubly transitive on the other 6 circles.

8.

The higher dimensional cases.,

For t a positive integer, let

be a spread (this time, of (t-1)~dimensionalprojective subspaces) of PG(2t such

d

- 1,q).

Just a s in section 7 (which corresponds t o t = 2) t o each

there corresponds a translation plane .rr(J)

of order qt.

How-

ever, for t > 2, the theory, t o my mind, i s much more rudimentary. For t = 2, the reguli of a doubly ruled quadric in PG(3, q) played an important role.

I looked for a suitable analog of this quadric in

PG(2t - l,q), and found it for the case that t is an odd prime.

The

corresponding surfaces are algebraic surfaces of degree t in PG(2t

- 1,q)

and are ruled by t distinct classes of (t-1)-dimensional projective subspaces.

These surfaces (or hypersurfaces, a s one auditor insisted) seem

R. H. Bruck

t o b e quite new.

See Bruck [4].

A regulus of PG(2t

-

1,q) is a s e t

&

of q

+1

disjoint

(t- 1)-dimensional projective subspaces having t h e property that every line of PG(2t

- 1,g)

which meets 3 distinct spaces of

&

meets a l l

of them.

Any s e t of three disjoint (t-1)-spaces l i e s i n one and only one

regulus.

A spread

.d (of

(t-1)-spaces) is called regular *robided that,

d,d c o n t a i n s the corresponding regulus. i s regular precisely when is Desarguesian. ) If d is regu-

for every 3 diitinct spaces of (

$

-I{($)

lar, the system consisting of t h e spaces of and the reguli of

J,considered a s circles,

higher-dimensional "circle geometry".

d , considered a s points, is clearly some kind of

I have studied this circle-

geometry in some detail, again for the c a s e that t is an odd prime.

(See

Bruck [ 5 , 6 ] . ) I t should be remarked that the spreads of the present section are quite different from the spreads of lines of PG(2t were discussed briefly in section 5.

- l,q),

t > 2, which

The first seven sections of the

paper form a unified whole, but the present section is related only t o section 7.

R. H. Bruck Bibliography 1.

J. ~ n d r e ' ,Uber nicht-Desarguesche Ebenen mit transitiver

Translations gruppe. 2.

Math. Zeitschr. 60, 156- 186 (1954).

R H. Bruck, Existence problems for c l a s s e s of finite projec-

tive planes.

Unpublished lecture notes of the 1963 Summer Conference

of the Canadian Mathematical Congress i n Saskatoon 3.

R H. Bruck, Construction problems of finite projective planes.

Combinatorial Mathematics and its Applications (Proceedings of the 1967 Chapel Hill Conference. ) Chapter 27, 426-514.

University of North

Carolina Press, 1969. 4.

R H. Bruck, Some relatively unknown ruled surfaces i n pro-

jective space.

Archives, Nouvelle Serie.

Section d e s Sciences,

Institut Grand-Ducal d e Luxembourg, 34, 361-376 (1974). 5.

R H. Bruck, Circle geometry i n higher dimensions.

(To appear

i n a birthday volume for R C. Bose. ) 6.

R H. Bruck, Circle geometry in higher dimensions, 11.

Geometrial Dedicata (to appear). 7.

R H. Bruck and R C. Bose, The construction of translation

planes from projective spaces. 8.

R. H. Bruck and R C. Bose, Linear representations of projec-

tive planes i n projective spaces. 9.

44.

4, 117-172 (1966).

Ergebnisse der Mathematik

Springer-Verlag, New York, 1968.

H. LUneburg, Die Suzukigruppen und ihre Geometrien

Notes in Mathematics. 1965.

Journal of Algebra,

P. Dembowski, Finite Geometries.

und ihrer Grenzgebiete, 10.

Journal of Algebra, 1, 85-102 (1964).

Lecture

Springer-Verlag, Berlin, Heidelberg, New York,

R. H. Bruck 13. Esther Seiden, On a geometrical method of construction of partially balanced designs with two associate classes, Annals of Mathematical Statistics 32. 1177-1 180 (1961). 14. R H. F. Denniston, Some packings of projective space, Lincei Rendiconti (to appear).

J. A. Thas

Flocks of Finite Egglike Inversive Planes

L

A. Thas

An w o i d 0 of the threedimensional projective h 2 space PG(3,q), q = p and q > 2, is a s e t of . q 1 points no three of 1.

INTRODUCTION.

+

which are collinear. If 0 is a n ovoid of PG(3,q), then a n incidence structure I(0) = (0, B, E) is defined a s follows: (i)

Points are the elements of 0;

(ii)

Blocks are called circles and are the s e t s P n 0, where P

is a plane of PG(3,q) with 10 n PI > 1. I t i s straightforward t o prove that I(0) is a n inversive plane of order q. We c a l l a finite inversive plane egglike if it is isomorphic t o a n I(0) for some ovoid 0.

We remark that every finite inversive plane of

even order is egglike [3]. A flock of a finite inversive plane I is a s e t a of mutually dis-

joint circles of I such that, with the exception of precisely two points x and y, every point of I is on a (necessarily unique) circle of a.

The

points x, y are called the carriers of the flock. Consider a n egglike inversive plane I(0) (0 is a n ovoid of PG(3, q)) and l e t L be a line of PG(3, q) which has no point in common with 0.

Then the circles P n 0, where P is a plane containing L with

1

IP n 0 > 1, form a flock of I(0). It was conjectured that every flock of I(0) could be obtained in that way ([I], [ Z ] ) .

In t h i s paper we prove that

the conjecture i s true i n the case that q is e v e n

J.. A. Thas

2.

LEMMA.

If L is a s e t of q t 1 .points i n PG(3,q), which has a

non-empty intersection with every plane, then L is a line of PG(3,q). Proof. -

Let L = {xl,x2,.

..,xq+1)

and suppose that L is not a

Then the line xlxZ = Lf contains a point y, with y # L.

line.

be a line such that y

e

Lw and L1In L = 9.

Let L"

Since e a c h plane through L1'

I L I = q + 1,

9 and

contains a t l e a s t one point of L and since Ln n L =

we conclude that every plane through Lw contains exactly one point of L. As

IL

I

So we conclude that

n plane LfLIt _> 2, we obtain a contradiction.

L is a line. THEOREM.

3.

If

{c1,C2,.

...

Cq-l),

q = 2h and h > 1, is a flock of

t h e finite egglike inversive plane I(O), then the planes of the q

-1

circles Ci a l l pass through t h e same line L This line L is the intersection of the tangent planes of 0 a t the carriers x, y of the flock.

...u

Proof. -

First of a l l we remark that 0 = CIU CZu

..,nq- 1)

is a line.

.

Now we shall prove that

The nucleus of the circle Ci is denoted by ni. L = {x, y, nl,.

Cq-lU 1x3 U {y)

For that purpose we show that every

plane of PG(3,q) has a t l e a s t one point in common with L. a)

Every plane through x or y has a point i n common with L.

b)

Let P be t h e tangent plane of 0 a t p

Through p there passes a circle of the flock, s a y Ci.

c

0 (p # x, p

Let P be a plane for which P n 0

{c1, CZ...

c

y).

The tangent line

of Ci a t p is contained in P, and s o ni c P. c)

#

..Cq-l).

If Ci = P n 0, then ni e P. d) Y

# P, P n 0 /

AS

Finally l e t P be a plane for which

,...,

{c1

Cq-l).

q t 1 is odd there exists a C

IP

Ici n C I=

If P n 0 = C, then j

such that IC. I

n 0 I > 1 and x / P, n C ] r {0,1,2}. 1.

There follows

J . A. Thas

that C . and C have a common tangent line T a t their common point. ,

1

And s o n. c T C P, from which n. r P. I

I

We conclude that every plane of PG(3,q) has a t l e a s t one point in common with the s e t L of order q

+ 1.

From the lemma there folloG3

that L is a line of PG(3, q). Next we remark that the polar planes of x, y, nl,. respect t o the symplectic polarity

7~

..,n9-1' '

with

defined by 0, are the tangent plane

of 0 a t x, the tangent plane of 0 a t y, and the planes of the circles Ci0 A s L is a line, these q of L with respect t o n.

+1

planes a l l pass through the polar line

So we conclude that the planes of the q

-1

circles Ci a l l p a s s through the intersection of the tangent planes of 0 a t the carriers x, y of the flock. BIBILIOGRAPHY [l]

R H. BRUCK, Finite geometric structures and their applications:, Construction of finite planes, 2nd C. L M. E. Session 1972 (Bres sanone).

[2]

P. DEMBOWSKI, "Finite geometriesg1,Springer-Verlag,

1968,

275 pp. [3]

P. DEMBOWSKI and D. R, HUGHES, On finite inversive planes,

J. London M a t h Soc.

, 40

(1965), 171-182. Prof. Dr. Je A. Thas Seminar of Higher Geometry University of Ghent J. Plateaustraat 22 9000 GENT BELGIUM

C E N T R O INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E .

R. H. F. DENNISTON

Corso tenuto

a

Bressanone

dal

18 a1 2 7

Giugno

1972

PACKINGS OF PG(3,q) by

R. H. F .

Denniston

(University of L e i c e s t e r )

One of the conjectures discussed in Bruckls lectures is the pos-

+

sibility of constructing a projective plane of o r d e r q(q

I), where

q

would be s o m e power of a prime. The l l p o i n t s l l would b e a l l the points, together with s o m e s p r e a d s of lines, i n PG(3, q): the "lines" a l l the lines of the space,

would be

together with s o m e suitably chosen packings.

(See the lecture notes f o r definitions). This construction would be impossible if t h e r e w e r e no packings

- but

in fact many different

packings do exist. In PG(3, 2) , packings fall into two transitivity c l a s s e s under the collineation group (Cole, Bull. Amer. Math. Soc. , 28 (1922) 435 -7). These two c l a s s e s are, however, interchanged by any duality (the concept "packing" all packings of

being regarded a s self-dual); and s o we may say that

PG(3,2) a r e alike. The words "alike"

and "different",

in what follows, a r e to be understood in this strong sense. I have to t e l l you that, in every

PG(3, q) with

q

2, packings exist and a r e not all alike. A construction, all such values of

q,

g r e a t e r than effective f o r

is given in a note (now in the p r e s s f o r Rend.

Accad. Lincei) : I give a short description of it, and show one way of adapting it to furnish a second packing. By Kleinls method, the points of sent the linear complexes of

PG(5, q)

a r e used to r e p r e -

PG(3, q). A complicated construction in

the higher space f u r n i s h e ~a regular s p r e a d & of 2 q + q other regular spreads ,f2, . . ; and

Yl,

each

i,

is a

regulus

9.. We 1

switch

lectures) to give a subregular spread

.

!f?,i

out of of index

PG(3, q),

J% n $ i,

yi

and for

( s e e Bruckls

1 -and a l -

R. H. F. Denniston s o switch

q out of

J% to give a spread

i

. My note proves that

is a packing: and a different packing is

I have done some work on PG(3,3), and found a packing that consists (like those just mentioned) of one regular spread and the r e s t subregular, but is m o r e symmetrical than they a r e . Then there were three different packings consisting entirely of subregular spreads; and three more, in each of which all but two of the spreads were regular. So PG(3,3), where the spreads a r e of only two different types, has packings of at least nine; I image that the number of types is considerably m o r e than 9 when q = 3, and that it increases rapidly with q

.

Let u s define a "cyclic packing" a s a s e t of tq2

+

+

1 disjo2 int spreads, permuted cyclically by a collineation of period q + q + 1 .

The packing of

PG(3,2)

q

is in fact cyclic; and the concept is di-

scussed generally by C. R. Rao (Proceedings of the 1967 Chapel Hill Conference). There is no cyclic packing of

PG(3,4)

because any collineation

of period 21 has some line-orbits of lengthless than 21; and I should 2 expect the s a m e difficulty to a r i s e whenever q + q + 1 is not a p r i me. I have also found, by searching, that there is no cyclic packing of

PG(3,3). Curiously enough, there is a BIB design,

with a cyclic

packing, which has the s a m e parameters a s the design of points and lines in

PG(3, 3)

(Moore, Amer. J. Math. 18 (1896) 264-303).

At the end of this talk, I give a short arithmetical specification

R . H. F . Denniston

of six different cyclic packings of

PG(3, 8). One of these consists en-

tirely of regular spreads; and s o does the packing of spread in that space being regular. But, in

PG(3,2), every

PG(3,3), I have c a r r i e d

out a complete s e a r c h by sorting punched cards, and found that no packing with regular spreads exists. So we may take some interest in two unsolved problems: "Which three-dimensional spaces have cyclic packings?",

and "Which spaces have

packings that consist entcrely of r e -

gular spreads? ". We could get another unsolved problem by making a definition: 2 A "complete partial packing of deficiency d" is a s e t of q + q + 1 disjoint spreads in

-d

PG(3, q), such that no other spread can be found

that is disjoint f r o m all of these. Mesner (Can. J. Math. 19 (1967) 273-280) has made an analogous definition of "complete partial spread", and has set up, in connection with his definition, a substantial lower bound for the (positive) deficiency. I have found, in

PG(3, 3 ) ,

a com-

plete partial packing of deficiency 2 : on the other hand, a deficiency of 1 is easily s e e n to be impossible. Does the minimum deficiency inc r e a s e a s a function of q ? But the important unsolved problem is: "Can Bruckls hypothet ical construction e v e r be carried out?" About that I have nothing much to report, except f o r the variety of packings shown to exist, which does seem encouraging. In any s e t of packings that satisfied all the conditions f o r the construction, only a v e r y few (if any) could be of the type given by the process that I can c a r r y out in a general

PG(3, q).

The next step, after finding a packing , P, should be t o find what we might call "a transversal?' od

P

-a

spread in which no

two lines belong to s a m e spread of P. Suppose, in fact, that Bruck's construction has been carried out, and that which a r e used a s "lines"

is one of the packings

of the plane. Then any two lines, if they a r e

skew and belong to different spreads of s v e r s a l of

P

P, must belong to some t r a n -

P. F o r each of the packings I have found in

PG(3,3), I

R. H. F. Denniston

have easily

verified that t h e r e a r e not enough transversals to satisfy

the condition just mentioned. In

PG(3,4), the packing given by my

f i r s t construction has many transversals, but t h e r e a r e other conditions they do not satisfy. In

PG(3,8),

on the other hand, I have s o f a r not managed to

find any transversal to any of the packings discovered. This must be regarded a s a failure to think of a suitable method, since some transversals presumably exist; in fact, 72 s e e m s the least unlikely o r d e r for a projective plane constructed by Bruck's method, and accordingly I hope that somebody will be able to continue my work. ~ ~ ( 3 , ' 8a)r e pu-

Since it will be some t i m e before the results in

blished, I give h e r e enough arithmetic to enable the packings to be r e constructed by anyone who is interested. Let i

3 GF(2 )

such that

be constructed by adjoining to G F ( 2 ) an element

i3 = i2 + 1

nates over this field, and

.

Let

(x, y, z)

be non-homogeneous coordi-

let

g be the line y = 0, 0 collineations, with the respective periods 73, 7, 9,

T : (x,y,z)

+ (i3x + i6y + i z ,

U : (x, y, z)

-+ (ix, iy, iz),

L e t numbers

tO,

. . .,t6

5

3

3

i x + i y + i z,

3

z = 1.

Let

be specified by 5

i x + i y ' + i z),

be ,given by any row of the following t a -

ble ( a complete table would have twenty rows, but the packings s o spe,'

cified would not all be different) :

R. H. F. Denniston

3

Let

be a s e t of 65 lines, two of which a r e the z-axis and

the line at infinity on

z = 0,

while the others a r e

Then i t wiU be found that

If.

TQ

.

~ ~ ,..., 8 '

.

Rows

(c)

and

is a spread, and moreover that

is a packing. Row

the table gives a regular spread of index 1

8

(d)

( a ) of

, and (b) a subregular spread give two different s p r e a d s which a r e

not subregular, but each of which contains seven disjoint reguli. Rows (e)

and

(f)

give two different spreads, neither containing any regulus.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E . )

J. DOYEN

RECENT RESULTS ON STEINER T R I P L E SYSTEMS

C o r s o tenuto a B r e s s a n o n e d a l 18 a1 2 7 G i u g n o 1 9 7 2

Fl3CENT RESULTS ON STEINER TRIPLE SYSTEElS Jean DOYEN Department of Mathematics University of Brussels 1050 Brussels, Selgium

1. Introduction. A Steiner trinle system (briefly ~ T S ) oC order v is a finite non-empty set S of v elements (called

2oints), together with a colLection of subsets of S (called lines) such that every line has exactly 3 points and every pair of goints is contained in exactly one line ; a Steiner triple

cystern of order v is sometime8 denoted simply by ~ ( v ) .For

) c x a ~ p l ,e any d-dimensional finite pr jective space over G F ( ~ ic ax? ~ ( 2 ~ " - 1) and any d-dimensional finite affine space over d ~ " ( 7 )is a n ~ ( ). 3

An ~ ( v ) is nothing else than a balanced incomplete block desif~llwith parameters

A=

v, b

= v(v

-

1)/6, r = ( v

- 1)/2,

k = 3,

I. Ae b and r must be integers, it follows, that v H 1 or 7

( ~ o d6). ~ i r k m a n p 3 ]proved in 1847 that this necessary condition of exi6tence.i~also euflicient : in other words, there is an ~ ( v ) for every

v s l or 3 (mod 6).

2. Constructions. Roughly speaking, the methods of cons-

tion of 9TS are of two types : the direct

construction^, in

which an STS is constructed directly Sroln nn algebraic structure, and the recursive constructions, in which an ST8 is obtained from a collection of "smaller" STS. ICe shall give an exaqgle of e ~ c htype: (A)

ordsr 7t

Let G be a finite multiplicative abelian Proup of odd 4

1. T s ' w s s yoizts the 6t

+ 3

elements of the set

J. Doyen S = GX[O,I,~)

i

e.nd a s l i n e s t h e following s u b s e t s of 3 : x

0

1

2 f o r every

x G

.

,

( i 5 ) { ~ X , O ) ~ ( Y , O ) , ( ~{ ~( X ~ ,) ~) ) ~ ( Y , I ) , ( Z , ~ ) ] ~ ( x , ~ ) , ( Y , ~ ) , ( B ,foo )r ) every x f y

and

x, y ,

5

e G such t h s t

2

x y = 5 .

This d i r e c t c o n s t r u c t i o n , which i s e s s e n t i a l l y due t o ~ o s e c ? ] ,

+

y i e l d s a S t e i n e r t r i p l e system of o r d e r 6 t

3 on t h e s e t 3.

It can be ahown[6]that two t r i p l e S G' a r e i - s o ~ o r p h i ci f and o n l y i f t h e groups GI,

We s h a l l denote i t by

9 , S G1 G2 G2 a r e isomorphic. systems

( B ) L e t S1 = {xl,.

S2 =

I ...,%) yl,

.. m3 be

a t r i p l e system of o r d e r m and

,X

e t r i p l e system of o r d e r n. Take as p o i n t s t h e

mn elements of t h e product s e t

S = S XS2

and a s l i n e s t h e

1

f o l l o w i n g s u b s e t s of S :

( ~ ) ~ ( X ~ , ~ ~ ) . ( X ~ . Y ~ ) , ( Xf~o ,r Ye~v)e r)r ~ o i n x t i of every l i n e Err.Y, ,yt) of S2

s1

and

( ~ i ) { ( x ~ , y ~ ) . ( x ~ , y ~ ) , ( ~f~o~r ~every )) l i n e {xi,xj,%) and every p o i n t y (xj.ys).

(

~

r ~

of

of S2 9

f~ o r~ every 1 ) l i n e (xi*xj*\\

of

S and every l i n e {yr, rs .yt) of S2. 1 This r e c u r s i v e c o n s t r u c t i o n y i e l d s a S t e i n e r t r i p l e systepl of order

mn on t h e s e t 9. We s h a l l c a l l t h i s system t h e d i r e c t

product of S1, S2 and d e n o t e it simply by

H . Werner [ e P r o v e d

Y

X S2.

I

t h a t S t e i n e r t r i p l e systems have a

unique f a c t o r i a a t i o n p r o p e r t y with r e s p e c t t o t h i s product : t h e decomposition of a given 9TS i n t o i r r e d u c i b l e f a c t o r e i s unique up t o ieomorphim o f t h e f a c t o r s and up t o t h e i r o r d e r .

3. Ieomor~hisme. It i s a t i l l an open problem t o determine t h e number ~ ( v of ) pairwiqg-non isomorphic STS of a given o r d e r v . It i s easy t o check t h a t

~ ( 1= )~ ( 3 =)

De ~ a s ~ u a l e [ h ] ~ r o v e idn 1899 t h a t

~ ( 7 =)~

( 9 =) 1.

~ ( 1 3 )= 2. One of t h e ~ ( 1 3 )

.J. Doyen can be constructed by taking. a s p o i n t s the elements of t h e a d d i t i v e group

of i n t e g e r s modulo 13 and a s l i n e s a l l

Z,, AJ

subset. where line6

of t h e f o m { x ,

of Z

10,1,,1 x€

x

+

1, x

9

{0,2,7)

9

{2,4,9)

9

system and r e p l a c i n g them by [0,1,7) {2,7,9)

.

+

4) ,{x,

x

+

2, x

13 t h e o t h e r ~ ( 1 3 i) s obtained by removing t h e Z,,; {1,7,9)

,

+

73

from t h e above

20.2.4)

, 11,4,9)

.

The oomputation of ~ ( 1 5 was ) fi'ret attacked (by hand)

i n 1917 by Cole, Cwnminga and White

13):

they found

~ ( 1 5 )= 80.

A computer search c a r r i e d out i n 1955 by H a l l and ~wiftfi%]led t o the same concluLion. The values of ~ ( v f)o r

v 3 1 9 are

s t i l l unknown. The beet estimate of t h i s f u n c t i o n is due t o R. M. Wilson (unpublished ) : he proved t h a t 2 E (log v 5)

LP

f o r every

v f1 or

-

3 (mod 6 ) . noreover, i f van d e r 'nlaerden's

conjecture on t h e permanent of doubly s t o c h a o t i c m a t r i c e s i s t r u e ( s e e f o r instancee0)for f u r t h e r d e t a i l s ) , then t h e 12 i n t h e above lower bound can be replaced by a 6. Wilsonls lower bound i s n o t very good f o r m a l l values of v. However, applying h i e b a s i c i d e a t o t h e STS of o r d e r 19

and 21, one g e t s

~ ( 1 9 ) 3 8 8 9 4 and

N(21)>,2.10

6

.

4. Automor~hisme. Let us d e f i n e an incomplete S t e i n e r t r i p l e system a s a f i n i t e non-empty s e t S of v n o i n t s , t o g e t h e r with a c o l l e c t i o n of subsets of S, c a l l e d l i n e s , such t h a t every l i n e h a s e x a c t l y 3 pointe and every p a i r of p o i n t s i s contained i n a t n o s t one l i n e .

C . ~ r e a s h k q h a sproved t h a t

every incomplete STS can be ( f i n i t e l y ) completed i n t o an STS.

It i s not very d i f f i c u l t t o show [5] t h a t given a f i n i t e a b s t r a c t group G , t h e r e e x i s t s an incampleto STS whose autonorphiem group i e isomorphic t o G.However, t h e following problem i s s t i l l open:

J. Doyen Given a finite abetract group G, doee there exiet an STS whose automorphiwn group is isomorphic to G ? Another problem hae gained interest in the past few years: Given a permutation 0( on a set of cardinality v r l or 3 (mod 6), does there exiet an STS of order v admitting d as an automorphfsm ? We shall denote such a eystem by

~~(v).

If # has s.single cycle of length v, R. Peltesohn proved in 1939 that there is an S (v) for every (mod 6 ) , except for

a

n?]

v I1 or 3

o = 9.

If0< has just one fixed point and a cycle of length

v

-1

on the remaining pointe, A. Rosa (unpublished) proved that there is an so((v) if and only if If

O(

v f 3 or 9 (mod 24).

is an involution with only one fixed point, a neces-

eary and sufficient condition for the existence of an S (v) is o(

v l l , 3, 9 or 19 (mod 24) ( ~ o s a E a~oyenc~], ~ ~eirlinck@q). If O( is an involution with exactly 3 (necessarily collinear) fixed pointe, the existence of an S ( v ) has been conjectured for every

v s 1 or 3 (mod 6), v

constructed easily for every

+ 1u. Such systems can be

v s 3 (mod 6) (take one of the

systeme SG deecribed in section 2 (A) and consider the automor-1 ghism %of SG defined by O((x,i) = (x ,i) for every x e G and

i = 0,1,2), but the problem is still unsolved for

vJ1

(mod 6).

5. Subsustem~. ( ~ r o mnow on, we shall say that a positive integer v is admiosible if

vSl or 3 (nod 6))

.

A subset S 1 of a Steiner triple eystem S is called

R

subeyetern of S if every line of S joining two points of S' is entirely contained in S1. A subsystem S 1 is said to be of order v 1 if it has cardinality v'.

Clearly, any intersection of sub-

systems of S is again a subs~~stem of S, so that it makes sense to speak of the subsystem generated by any given subset of S.

J. Doyen If an ~ ( v ) containa e eubsystem S" of order v 1 easy to see that

v>,2v8

+

+ v,

it is

1. Conversely, Doyen and Wilson

bg

have proved that given a Steiner triple system 3' of order v1 and an admieeible integer v)2v1

+

1, there exists an S(4.

containing a srtbsyetem ieomorphic to 3'. Let S be a Steiner triple system of order v a 7 . A subeet T of S consisting of 3 non collinear pointe will bq .cgl-leda triangle of S. According to a classification introdri.c&a by

L. Snwnko+owicn

PI,S ie called

(i) a non deaenerated ~ l e m eif every trimgle of S generates the whole system (in other words, if S hae no proper embeystem), (ii) a degenerated

lane if one of the trianglesof S

generates the whole system and another one does not, ( i i i ) a swace if no triangle of S generates the whole

system. One can proveL6][~]that

there is a non degenerated plane

of order v for every admissible

-737 &d

a degenerated plane

of order v for every admissible ~ 3 1 5 .A; J. W. Hilton (unpublished) has constructed a space of order v for every admissible

v>867

but not much is known for smaller valuea of v.

For exaqple, the only admissible integers space of order v is known to exist are 6:,

~ ( 1 0 0 for which a

15, 27, 31, 45, 49, 55,

81, 91, 93 and 99. It is easy to see that there are no

spaces of order 19, 21 and 25. The existence of a space of order 33 is still in doubt. On the other hand, the only known examples of spaces in which all subsystems generated by triangles have the same order v 1 are those for which

v 1 = 7 or 9. It is tempting to

conjecture that there are no others. Note that if every trianzle. generates an ~ ( 7 the ) ~system is necessarily a projective

J. Doyen speoe ~~(d.2).However, if every triangle generatee an S(9), the system is not neoesearily an affine epaoe AG(d,3) :

M. ~all(??.haeoonetruoted an ~ ( 8 1 which ) is not ieamorphio to ~ ~ ( 4 but ~ 3 in ) which every triangle generates an ~ ( 9 );

...

s ( ~ ~ ) x s ( ~ ) x s ( ~ ) %% ~ ( 3 )

obviouely, any direct produot

givee an ~ ( 3 with ~ ) the same property for every

1134.

6. Disjoint STa. Two Steiner triple systems having the same eet of pointe are called disjoint if they have no line in common. . Let us denote by D(V) the maximum number of pairwise disjoint ~ ( v ) that can be oonetruoted on a given set S of v pointe (va3). As S contains

v(v

lity 3 and as any ~ ( v ) has tely

&I+]

- 1)(v - 2)/6 eubeek* of cardinav(v - 1)/6 lines, we have immedia-

- 2. Obviously,

~ ( 3 =) 1. Caylpy [sl and Kirkman proved in 1850 that ~ ( 7 = ) 2 and ~ ( 9 ) = 7. R. H. F. D(v)S v

Denniston (unpublished) has ehown with a computer that ~ ( 1 3 )= 11

,

~ ( 1 5 )is not yet known. A lower bound

but the value of

for ~ ( v ) can be found in L9:g'. It has been conjectured that admissible ~ ( v ) = v

~(v) = v

-2

for every

v) 9. 'L. Teirlinck c ~ ~ ~ 1 ~ r orecently ved thgt if

- 2, then

D(3v)

3

3v

- 2.

This, together with the

results mentioned above, implies that the function ~ ( v ) achieves its maximum value for every

v of the form

3n or 13.3".

Two disjoint Steiner triple eyetema S1, S2 are said to'be orthosonal if whenever two pairs of poits appear with the same third point in lines of S1, they a2pear with distinct third pointe in lines of S2.

.'

A pair of orthogonal ST3 of order v has been constmated .

for infinitely many values of and

N. S.Kendelsohn

v.tl (mod 6)

by C. C. Lindner

1-153 . On the other hand,

the non exietence

J. Doyen of a pair of orthogonal ST9 of order v ie obvioue for and has been establiehed by R.C

.

Hullin and E

v = 3

. Nemeth fiqfor

v = 9. There was a conjecture that such a p e r does not exiet

for

v *3 (mod 61, but Rose [

has exhibited recently two orthogonal STS of order 27. Not much ie known for the other values of v.

1. R.C.Boee

: On the conetruotion of balanced incomplete

block designs, Ann: Enaenioe 9 (1939)~353-399. 2. A. Cayley : On the triadic arrangements of eeven and fifteen things, London. Edinburgh and Dublin Philoe. 11ae. and J. Soi. (3) 37 (1850)~50-53. 3. F.N.Cole, L.D.Cummings and H.S.White

: The complete enume-

ration of triad eyeteme in 15 elemente, Proc. Nat. AcadL Sci. U.S.4.

3 (1917), 197-199.

4. V. de Paequale : Sui eietemi ternari di 13 elementi, Rend. R. 1st. Lombardo Sci. e ~ett.(2) 32 (1899), 213-221.

5. J. Doyen

: Constructions groupales d'eepacee

linbairee

finis, Acad. ROY, Bele. Bull. C1. Sci. 54 (1968), 144-156. 6. J. Doyen : Sur la etructure de certains syetemee triples de Steiner, Math. Zeitechr. Ill (1969), 289-300.

7. J. Doyen : Systemes triples de Steiner non engendres par tous leurs triangles, Math. Zeitechr. 118 (1970). 197-206. 8. J. Doyen : A note on reverse Steiner triple systems,

Discrete Math. 1 (1972). 315-319.

9. J. Doyen : Constructione of disjoint Steiner triple eystems, Proc. Amer. Piath. Soc. 32 (1972). 409-416. 10. J. Doyen and R.N.

1;ilson : Embeddings of Steiner triple

syeteme, Discrete Aath. (to appear). 11. M. Hall, Jr. : Automorphiame of Steiner triple eystems, I 5 N J. Res. Develo~. 4 (1960), 460-472.

J. Doyen 12. M. Hall, Jr. and J.D.Swift

: Determination of Steiner

triple eysteme of order 15, EIath. Tables Aids CQDIUU~.9 (1955 ) 9 146-156. 13. T.?.Kirkman

: On a problem in combinations, Cambridae and

Dublin Math. J. 2 (1847). 191-204. '14.

T.P.Eirknan

: Note on azv ~~nenawered ~ r i e equestion,

Cambridve and 3 u b l i n Math. J. 5 (1850), 255-262. 15. C.C.Lini?ner and N.S.nlendelsohn

: Construction '30

cular Steiner quasigroups (to appear)

.

aerpendi-

1-6. R.C.Mullin d E.Nemeth : On the nonexiutence of orthogonal Steiner systems of order 9, Canad. Iath. Bull. 13 (19701, 131-134. 17. R.Peltesohn : Eine Lasung der beiden Heffterechen Differenaenprobleme, Comaoeitio Math. 6 (1939)~251-257. 18. A.Roea

: On reverse Steiner t r j ~ l eeyetema, Diacrete Math.

2 (1972 ) , 61-71. 19. A.Rosa

: On the falsity of a conjecture on orthogonal

Steiner triple systems (to appear). 20. H. J .Ryeer : Combinetorial Fiathematics ( c a m e Elath. Monograph No. 14), Wiley, New York 1963. 21. L. Saamko3owica : Sur une claseificntion des triplets de Steiner, Rend. Accad. Nan. Lincei (8)36 (1964)~325-128. 22. L. Tei~linck: On the maximum number of disjoint Steiner

triple eysteme (to appear).

23. L. Teirlinck : The existence of reverse Steiner triple systems (to appear). 24. C.A.Treash

: The completion of finite incomplete Bteiner

triple systems with applications to loop theory, J. Combinat. Theory 25. iI.Werner

Yer. A 10 (1971), 259-265.

: A unique factoriziation theorem for Steiner

triple systems (to appear)

.

CENTRO INTERNAZIONALE MATEMA TIC0 ESTIVO (C. I. M. E. )

GRUPPEN UND ENDLICHE

Corso tenuto

a

Bressanone

dal

PROJEKTIVE EBENEN

1 8 a1 2 7

Giugno

1972

~~PP,~~=~P,~~~~~~~~PZPJ~E~~!!~=ES:,~!! von Heins Liineburg

Ein wichtiger Teil der Theorie der endlichen projektiven Ebenen besteht in der Charakterisierung bekannter Ebenen durch ihre Kollineationsgruppen bm. durch Untergruppen ihrer Kollineationsgruppen. Bei solchen Charakterisierungen mu13 man sich nun in der Regel der verschiedenartigsten Hilfsmittel bedienen. Dies sol1 hier am Beispiel der von 0. Prohaaka stammenden Charakteriaierung der Hall-Ebenen detailliert beschrieben werden.

1. Projektive ug-affine menen. 1st

=P====-========E----PPP=========

eine M e w , deren Elemente

wir Punkte, und $ eine M e w , deren Elemente wir Blocke nennen, ist ferner I E 7r,

, so nennen wir das Tripe1 (P,g,1)

eine J n & -

denzstnrktur. In aewissen Spezialfallen, so z. B. bei projektiven und affinen Ebenen, nennen wir die Blocke auch Geraden un& drsetzen

den hchstaben

durch

9

.

1st P C

und

6

wir statt (P,4 ) C- I bzw. (P, 4-) 4 I meist P I

1st

6,

so heiBt

.$, so schreiben 4. bzw. P f 4e

.

,I) eine Inzidenzstruktur, deren Blocke wir Geraden nennen,

('P,Q ,I) eine pro.iektive Ebene,,falls (?i ,a ,I) die fo1-n-

den Bedin(yun(generfiillt:

1) Sind P,Q E P,Q 1

g.

und ist P

Q, so ~ i b tes @mu

3

ein g 12 $,

mit

Heinz Ltineburg

2) Sind g,h e

9 , so gibt es wenigstens ein P E

3) Es gibt vier verschiedene Punkte in Q

mit P I &he

, von denen keine drei

kollinear sind.

Dabei heiBen die Punkte P,Q,R,. g

9 gibt mit P,Q,R,...

.. E

kollinear, falls es ein

I s.

~ 7 ,a = ,I) heiQt affine Ebene, falls

,q ,I) die folenden Beding-

ungen erfitllt:

1') Sind P,Q 6 P,Q T

und ist P

4

Q, so gibt es genau ein

R

9

mit

s.

2') 1st P G

,g E 9

und ist P

1 g,

so gibt es genau ein h e

mit P I h und h n g = $. (~abeiist generell der Durchschnitt zweier Geraden g und h durch g n h = f ~ f X&

CY;! , X

3') Es gibt drei nicht kollineare Punkte in

I g,h\ definiert.)

.

Zwei Geraden einer affinen Ebene heiflen parallel, falls sie entweder gleich sind oder aber keinen Punkt gemeinsam haben. Sind g und h parallel, so schreiben wir g n h. Die Pamllelitatsrelation ist eine ~quivalenzrelation.Die Klassen dieser Relation nennen wir Parallelenscharen.

1st

=

@ ,(a ,I) eine projektive aerie und ist g

E

9

, so sei

Heinz Liineburg

Dann ist

'eine affine Ebene und man erhllt jede affine Fbene auf

diese Weise.

Es sei

,

(p ,% ,I) eine Inzidenzstruktur. 1st vd = $

Id = c ( 1 ,P) 1 (P, 6 ) C- 11, so ist auch itruktur. Sie heiI3t die xu (72 ,% ,I) bar ist

,

Inzidenzstruktur. Offen-

1st

=

(T,~,I) eine prajek-

= (yd,gd,Id) cine pmjektive Fbene.

tive Ebene, so ist auch

Die Inzidenzstruktur (1: ,$.,I) heiDt endlich, falls damit auch I endlich sind. 1st

wir v = 1 11:I und b =

'.

(7,%,I)

= rp und IIQI

I

und

d $d,Id ) eine Inzidenz-

(vdd,&dd,~dd) - (T,$-,I).

Dann ist lIpl = )I(P)(

=

I

und $ und

endlich, so setzen wir

- \1(6

.

) ( = k4 Ferner setzen Weil sovohl { I ~ t~ TP\ als auch

4 1 [~ i t - $ ( eine Partition von I k t , ~ i l t 1 .l.

Satz. Fiir jede endliche Inzi_denzstruktur(T ,%,I)

=====I==*=

Von besonderem Interesse sind die sog. taktischen K o n f i w r a t m , das sind diejenigen endlichen Inzidenzstrukturen, fiir die rp = r und k = k fiir alle P und alle

3-

R.

p i l t , Fiir taktische Konfiuratio-

nen hat man die Cleichung vr

1st

fi eine

-

bk, die unmittelbar aus 1.1

pro jektive Ebene und sind g und h braden

folgt.

V O E ~ so

6ibt es stets einen Punkt P, der weder auf g noch auf h liegt. (~iesbeweist man mit Hilfe von j).) t

Definiert man 6 durch

xG

I

W n h fiir alle X I g, so wird 5 rm einer Bijektinn der Menge

der M t e auf g auf die Menge der Punkte von h. Daher liegsn arf g ebensoviele Punkte wie auf h, Es ist ebenfalls leicht einzusehen, daB es zu jedem Punkt P und zu jeder Geraden g eine Bijektion der Menge der Geraden durch P auf die Menge der Punkte auf g gibt. Hieeine endliche projektive Ebene, so gibt es eine

raus folgt: 1st

natiirliche Zahl n ) 2, so daB auf jeder Geraden von Pwkte liegen und durch jeden Punkt g e m u n raus fole w e e n v(n

+

1) = b(n

+

genau n

+ 1 Geraden when.

den von P verschiedenen Punkten von

und die Inzidenzstruktur aus und den Geraden durch P,

so ist dies eine taktische Konfiguration rnit den Fametern I

1, b' = n

so da0 v = b = n2

1.2.

-

Satz. 1st

======i==i

+ 1.

+n+1

Hie-

1) weiter, daB v = b ist. Betrach-

tat man ferner einen Punkt P von

k' = n, r'

+1

Daher ist v

-1

- v'r'

Y'

b'k' = (n

=v

+

- 1,

l)n,

ist. Es gilt also

eine endliche ~ro~iektive Ebene. so aibt es eine

natiirlichex n 2, So daE folandes gilt: 2 1) besitzt n + n + 1 Punkte. 2)

besitzt n2

+n+

1 Geraden.

+ 1 Punkte. n + 1 Geraden.

3) Auf jeder Geraden lie~enn 4) Durch jeden Punkt 1st u m k e h r t

eine Inzidenzstruktur mit den Eigenschaften 1)

a

Heinc LUnebuq stet6 m i t m u

3) und inzidieren zwei verschiedene F'unkte von einer Geraden. eo ist

im Falle n

2

2 eine endliche wojektive

Ebene.

Benutzt man, da8 jede affine Ebene van der Form

1.2.

i e t , eo f o l g t

-

Satz. I e t A eine endliche a.ffine Ebene. so nibt ee eine natSir-

I i -111111

liche !hhl n & 2 , so da0 foluendee n i l t : 1) A besitzt n2 Funkte. 2 2) A besitzt n

+n

Geradan.

3) A u f .ieder Ceraden von A lie-

4) Durch jeden Punkt -hen n

n Funkte,

+ 1 Geraden.

5) Jade Farallelenecher e n t h u t n Geraden,

3) erflillt. und

1 s t w k e l u c t A eine I n z i d e n z s t ~ .die 1)

inzidieren zwei verschiedene Punkte von A s t e t s mit m u einer Geraden. so ist A eine affine Ebene. f a l l s nur n ),2

Die Zahl n aus 1.2 bzw. 1.3 heiBt die O r d n q der projektiven bzw. affinen Ebene.

,

Eine Inzidenzstruktur @ Q ,I) heiBt ein

Nets,

f a l l s s i e d i e f ol-

genden Bedingungen erfiillt :

1") Zwei verschiedene -Rmkte aus

Geraden aus

2") Zu P

p

inzidieren m i t hiochstens einer

91

t 7Q

und 8 E

CJ

mit P

#

g uibt es penau ein h 6

3mit

Heinz Liineburg

P I hund h n g =

a.

3") Es gibt drei nicht kollineare Punkte, die zu je zweien eine Verbindlmgegerade haben.

Jede affine mene ist also ein Netz und wie bei affinen Ebenen definieren wir auch auf den Geraden eines Netzes eine ParallelitBtere-

,

lation I/ die genau wie im F'alle der affinen Ebenen. eine iiquivelensrelation ist. Mit Rilfe von 2") und 3") ergibt sich ferner, daB jedes Netz eine taktische Konfiguration ist. 1st n die Punkteanzahl auf einer Geraden, so ist n auch die Anzahl der Geraden in einer

.

Parallelenschar, wie aus 2") folgt. Daher ist v = n2 Ferner ist rn = b, falls r die Bnzahl der Geraden durch einen Punkt ist. r heiBt gelegentlich auch der

des Netzes, warend n wiederum die Ordnung

des Netzes ist. Es gilt offenbar 3 s r s n + 1 und r = n

+1

genau

dam, wenn das Netz eine affine Ebene ist.

2. Translationeebenen. Isomorphismen von Inzidenzstrukturen definiert

......................

man in naheliegender Weise. Isomorphismen von projektiven bzw. affi-

nen Ebenen sowie von Netzen auf sich selbst nennen wir Kollineationen. -

Isomorphismen von

auf

heiBen DualitBten.

In der Theorie der projektiven Ebenen sind die axialen und zentralen Kollineationen von besonderem Interesse. Dabei heiBt die Kollineation G

axial mit der

Achse

g, falls O die Cerade g punktweise festla13t.

Zentral wird dual definiert. Es gilt der Satz, &a13 eine Kollineation genau dann axial ist, wenn sie zentral ist. Pian nennt axiale Kolli-

neationen daher auch -pektivitaten.

1st G axial rnit der Achse g

und dem Zentrum P, so heiDt Ci im Falle P f g eine Homolouie oder auch

eine Stmckimg und im Falle P I g eine Elation. Die Elationen rnit der Achse g bilden eine Untergruppe der Kollineationsgruppe (~eweia!)

-

,

die wir rnit ~ ( g )bezeichnen. Die Bemerkung, da0 die IdentitPt die einzige Perspektivitat rnit zwei verschiedenen Zentmn (~chsen)ist, liefert insbesondere,

~ ( g )auf der Menge der M t e von

?T

scharf

tranaitiv operiert, falls sie traneitiv operiert. Operiert ~ ( g )auf der Menge der Punkte von

transitiv, so heil3t

7-r

eine Translations-

ebene. T(~) wird In diesem Falle auch Translationsm~p%und die m e mente von ~ ( g )auch Translationen e;ensnnt.

Es sei

eine Translationsebene. 1st P I g, so bezeichnen wir rnit

T(P,~) die Untergruppe aller Elationen mit dern Zentrum P und der Achse R. 1st t E: ~(g), so ist ~"T(P,~)c

= T(Pc,g3

=

T(P,~), so da0

T(P,~) ein Normalteiler von ~ ( g )ist. Ferner ist ~ ( g )= U T(P,~)

7'13

und T(P,~)(\ T(Q,~)

a

{I$, falls P f Q ist. 1st G 6 T(P,~) und

r E T(Q,~), so ist, da die Gruppen T(x,~) ja Normalteiler von ~ ( g ) sind, cr" t-' nur P

#

cr t G T(P,~) n @,g),

Q ist. Gilt

ri,

so d&

sr

=

r

ist, falls

Z- E T(P,~) und ist Q f P, so ~ i b tes ein

9 C T(Q,~) rnit q f 1. Dann ist t y +T(P,U) und daher

so daB auch in diesem Falle

-

tcygilt. Daher haben wir

- 'eine Translationsebene. so ist ~ ( g )abelsch.

2.1. Sstz. 1st ==========

Setze

K(~) heiBt der Kern von ~(g). Nach AND&

/C43 let

~ ( g )ein Brper, so

daS ~ ( g )ein ~(g)-Vektorraum iat. Die T ( P , ~ ) sind UntemSume diesea \

Vektorraumea. Ferner gilt, dsS die Gruppe der Strec*. Achse g und Zentrum P

%

nit der

g mr multiplikativen Gruppe von ~ ( g )iso-

morph iet. SchlieSlich gilt noch, daB

'IT genau dann desargnessch

w e m T(~)als ~(g)-~ektorraumden Rang 2 hat (AND& loc. cit.). heiBt die projektive Ebene

ist,

hbei

deearguessch, wenn es einen Vektomum

V vom Range 3 iiber einem K6rper K gibt, so daB

&u

der In~idens-

struktur isomorph ist, die aus den Untedumen vom Renge 1 ale Punkten und den Unterraumen vom Rsnge 2 ale Geraden nit der Inklusion als Inzidenzrelation besteht.

-

-1 1

sei veiterhin eine Translationaebene. Ferner eei T

.ii=

~T(P,~)(PI 81. 1st dannn(T) = (T,txq(x t IT ,q eT),€),

so ist =(T) rm

~ ( g and )

isomorph, wie man mlihelos verifisiert. h.180-

mornhisma wird durch die Abbildung E 3 wenn 0 ein festgealter Punkt von

or

( Z_ GT) induziert,

ist. 1st andrerseite T eine

Gruope und iT eine nicht triviale Partition von T, dh. eine M e n g e von Untergruppen von T mit den Eigenschaften: a) Es ist T =

u X, X m

b) SindX,Y E r r und ist X Z Y, so ist X n Y = C 1 $ ,

c) TI-

enthat mindeatens zwei Untergruppen,

hat die Portition H ferner die Eigeneohaft, da8 fiir X,Y 6 r~ and X

4Y

s t e t s T = XY g i l t , so ist W(T) eine 'hmelationeebene und die

Abbildung r die g

*

T auf die durah xg = xg ( x € T) definierte Abbil-

dung g* abbildet, ist ein Isonorphismua von T out' die Gruppe der Translationen vm w(T). Insberrondere f o l g t nooh, deB T abelach ist. Eine nicht t r i v i a l e Partition mit dieser m & t % l i c h e nEigonnchaft, h e U t ~ ~ u a r t i t i von o nT.

2$2=fi~

(And&).

e b e Uwmenwartitiom der G r u ~ T, ~ e

Iat

ist V(T) eine !hamlationeebene und T* i e t die TranelationeR1P.DW

7

dieeer Ebene. &rf diese Wsise erhtilt man bia auf Isomomhie s l l e

Trsnelationsebsnan. Sind X,Y

i ' T , so gibt es ein

Z€ 7 m ~i t Z

/

X,Y, da eine Qhuppe

n i e d s Vereinigmg m i e r eahter Untergruppen i s t . h h e r bt T =

-xz

-m

una foigiich

Folglich sind a l l e Kom~onenlenvon T i isomorph. Dies g i l t sogar a l e

K(T)-~ektorraumisomorphi8mus, wenn K(T) die Menge der hdomorphismen w n T ist, die jede einzelne Komponente von g i n sich abbilden. K(T) i a t natiirlich mun Kern von T* isomorph. Hat T endlichen Rsng iiber K(T), so i s t Rg T

-

2n und Rg X

-

n fiir a l l e X € T

.

1st L(T)

der Verband der ~ n t e r r t i & ivon T, dh. die zu T gehSrige projektive Geometric, so ist a l s o

a eine tiberdeckung von

T m i t paaweis6 wind-

schiefen Untedumen dea Ranges n. 1st umppkehrt s~ eine b r d e c k u n g von 'P mit paarweise windechiefen Unterraumen dee Ranges n und i s t

Heinz Liimeburg Rg T = 2n, so ist TT eine Kongmenzpartition w n T. Diese projelttive

Betrachtungnweise der.Ibapuen%partitionen ist gele&entlich von NutZen.

1st T ein Vektorraum tom Range 2 iiber dem Kiirper K und ist

i~

die

Menge d m Untedume tom Range 1, so irt 7 eine Kongruenzpartibion

von T. Wegen K S. K(T) iet K = K(T),

g Sind

-

sa @, 9,I)

rind

60

daE T(T) desarguessch ist.

TO5 T:* %0 53 'lTO =

wO,go,~o)

U Q ~ IO

= 1

(Fo x go).

projektive Ebenen, so heiBt

5 eFna Ostembene von n. 1st no n, so heiBt & bane yon

0

n.

Untere-

ZZIZpg;(Bruok). z Ist & eine echte Unterebene der endlichen projektiven Ebene

TT,

& & m die OldrmnR yon

n, so ist entweder n

2

n oder m2

TTO & n die OrdmrnR von

+ m S n.

Dabei ist m2 = n ~leich-

bedeutend mit der Amawe: Jeder Punkt yon den von

lie&

auf einer Gera-

5.

Beweis. Es sei P ein Rmkt von

n, der mit keiner Gersden wn TTO

hzidiert. Dam enthYlt jede Gerade durch P hiichstens einen Funkt von

no.Weil andrerseits jeder Funkt yon noauf einer Geraden durch

P liegt, iatm2 + m + 1 6 n + 1, so daB in diesemFal1e m2 + m < n ist.

Jeder Funkt von Plmkt von

liege auf einer Geraden von

no.E8 sei P ein

T.Wegen m < n gibt es dann eine Gerade

g durch P, die

no,

den F'unkt P geme-am hat. Die m2 Gemden won die lit 2 nicht durch P gehen, schneiden g in m versohiedenen Punktcm, die

&

tibardies a l l e von P vemchiedan sin&.v e i l andreraeits jeder vtm g, der von P rerschieden i e t , auf @nau einer Geraden von

liegt, d i e dann notwendig zu jenen m2

-

n ist, q. e.

Im Falle m2

-

g &,

A

von

n.

eei eine projektive Ebme der Ordn~rdnungm2. Ferner

n. Ist g eine Gerade von &

y& B zwei vemchiedene Ptmkte von

und Rilt schlieI3lich

Beweis. Weil

6 Bae+Unterebene

Baer-Untembencm von

% y&

folgt, daB

a.

n heiBt

2 Hilfssatz. Zz=z==P==LII=II

d Geraden ,nMha,

no

{XIXFunkt

T o n n,

von

Tig ( i

-

TO,X I gt = { X ~ XPunkt

0,l) von

n1,

(die Definition dieaee hrrchechnitts ist die

naheliegende) vier Punkte e n t m l t , von denen keine drei kollinear sind, ist

qnnl eine Unterebene,

die auf Gnmd der wliteren Vor-

auasetzungen mindestena die O r d n u n g m hat. Folglich i s t

& l $ ~ % & g ~ : : ~ : = ~ ~ ~ ~ = ~ ~ ~ ~ E ~ ~ & = ~ ~bedmen ~ t = ~ ~ 2 % d g mit einem Hilfssatz.

&.l~;~fH~f~~z 7 eei eine endliche pmjektiw Ebene. Ferner s e i ( P , ~ )ein nicht inzidentes Punkt-Geraden.pear wn sei 8 5 -

~ ( g1 )14 1x1

=

q

+1a

n. SchlieBlich

t s e i eine zur ~ ~ ( 2 , qisomoh )

.

phe Kollinestionsmp~evon a)' P

b) C)

x+

Gilt dam:

= P,

='y* st x e Q , so ist ~(x,Px)= T(x,=) v

8

q, falls nur P # Q

1g

so ist ipl( LJ'Q

t-itiv

n

g \ix),

auf

die Arnktrnenae einer ~nteribeneder Ordrmng

8

FQ n g E

Gerriden durch P,

&Die

die rm dieser Unterebene neh6ren. sind die Geraden PX mit X C-

.

'Q

Beweis. Wir berechnen -2ichst

Z(X,PX) im Stabilisator & von Q in 1&\7,q

-

und folaich \Q'I

\t[/&l&q(q2

.

- 1)q-'

ist

=q2

-

1.

Weeen i 81 = q + 1 ), 3

8

gibt es zwei verschiedene Punkte X,Y rnit X,Y

und X,Y

Z(X,PX) mit hq PY ein

t$

.

enthalten. w e r iat

Es sei h eine Gerade durch P rnit ha g E f

c) gibt es zu jedem

P

W e e n X = FQ n g E

3

Z(Y,YP) mit

€

.

h. Ween

Q Z Ferner sei Ryq( = REG'.

hse' = h. Ea sei nun R I h und R

Hieraus folgt, daB R 4 , R W und Y kollinear sind. Da sndrerseita X, R9 und RW ebenfalls kollinear mind, folgt Q =

C~

-

Wegen \ $ I

\eZ\

3

-

q

+ 1 und

E(X,PT), hP# PYjI

p c

el. A180 ist \(RP"l

= q

- 1.

ist also

-

(q+ i)(q-I),

t aus

$

auf

der Transltivit=t von

so d a ~ \ ~ . Z \ =1qis ~t.

Duale Bchliisse zeigen: 1st h eine Gerade mit P ist eta M

und damit

G

g , so ist \ht \

diesem Falle die Relle von P und

3

= q2 =

- 1.

{PXIX

1h #

g und h n g

Dabei epielt g in

G8

1

die Rolle von

% Nun ist I < P ~ UQ Z V

gl

-

1

+ q2

- 1+q+1

=

2 q

+

q

+

1. Ferner

Reinz Liineburg

i s t (ig\ir h z

u 31 = q2

+q+

1. W l e n w i r h mm so, dafi auch

noch Q I h g i l t , so ist

m i t den Parametern v = b = q2

eine taktische Konfi-tion k = r = q

+ 1 und

+q+

der w i t e r e n Eiqenachaft, daB zwei Punkte von

suf hijchstens e i n e r Gersden von

noliegen.

Kieraue f o l g t , da6

1,

Ti0

5

eine Untembene d e r O r d n u n g q ist, q. e. d.

4t$zPi.ifggU2:442 Es s e i FG(j,q) d i e urojektive Geometrie d e r Dimen. sion 3 G g ( q

I G F ( ~ ) .Iat

-

3

e i n &mrboloill in FG(j,q),

112 Geraden von FG(),q),

e-

so ~ i b et s

nicht t n f f e n .

gg

+ q + 1 = (q + l ) ( q 2+I). 2 gleich (q2 + 1 ) (q +q +1),

Beweis. Die Punkteanzahl von FG(j,q) i s t q3 +q2 Daher i s t d i e Anzahl der Geraden von FG(3.q)

da durch zwei verschiedene Punkte genau eine Gerade geht. Weil e i n Hy-perboloid ist, enthiilt

ist d i e Anzahl d e r Geraden,

3 3 zwei Regelscharen ur~dQ2. Daher d i e gane i n 3 enthalten sind, g l e i c h

(y1(+1~2(=2(q+1).IstPC gi

G Qi m i t

3

P I gi.

liegen weitere q

In der Fbene gl

- 1 Geraden durch P.

,sogibtesenaueineGerade

+ g2

( ~ d d i t i o nim ~ e k t o r r a u m )

Diese haben m i t

meinsam und sind a l l e Geraden durch P, d i e m i t

'S tplr P qemeinsam

haben. Die r e s t l i c h e n q2 Geraden durch P t r e f f e n (q

- 2q - 1

+

-

qZ Funkten, d i e n i c h t auf g

Anzahl d e r Geraden, d i e m i t

+ 112(q - 1 ) und + 1)2q2 = v r = bk

a l s o (q aus (q

3 nur P Re-

1

3 gerade i n den + g2

l i e e n . Die

'5 nur einen Funkt gemeineam haben,

ist

d i e h z a h l b d e r Sekanten errechnet s i c h

-

2b zu

1 2 p (a + '

1)2, da d i e Punkte von

5

rmsaprmen mit d m Sekanten eine tcrktische Konfiguration bilden. Die Anzahl der Passanten iat also (q2+ 1)(q2

+ q + 1)

- 2(g

+

1)

- (q + ~ ) ~ ( -q 1) - F ( q

+

112.

- 02*

=

V aei ein Vektorraum vom Range 4 uber GF(~). Ferner seien Kongruenzpartitionen von V. 1st

;T

und rq

T(V) desarguessch (in diesem Falle

nennen wir TT eine desarguessche Kangruenzpartition), so ist GF(~)

im Kern K von TT(V) enthalten. Weil V iiber GF(~) den Raw 4 hat, 2 ist daher 2 und folglich K GF(~ ). Somit ist V ein

LK:GF(~)]

-

Vektorraum w m Range 2 iiber K. 1st

T'(v) ebenfalls desarmessch,

so ist V auch Vek:orraum vom Rang 2 uber den Kern K q von W(V) und 2 K' ist gleichfalls zu GF(~ ) isomorph. Somit gibt ea eine bijektive semilineare Abbildung O des K-Vektorraumes V auf den Kq-Vektorraum V. WeilTr gerade aus den UnterrYumen des Ranges 1 des K-Vektorraumes V und 7~

aus den Untedumen des Ranpes 1 des Kt-Vektorraumes V be-

steht, ist T T = ~ IT'.Nun ist GF(~) S K und GF(Q) G Kt. Weil ein endlicher Kiorper hochstens einen Kijrper gegebener O r d n u n g enthalt, folgt, daB 5 einen Automorphismus von GF(~) induziert. Somit ist Cr

eine semilineare Abbildung des GF(~)-~ektorraumes V auf sich,

welche TT auf 7~

Setzt man

-(r

=

abbildet.

m' in den vorstehenden Betrachtun(gen, so erhlilt

man,

daB die Gruppe aller semilinearen Abbildungen des K-Vektorraumes V

auf sich, die TT &variant laasen, zu ~ ~ ( 2 . q ~isomorph ) ist. Inagesamt erhalten wir, daB die Anzahl der desarguesschen Kongruenzpsrti-

1

tionen gleich \TL(4,q)\ /\\L(2,q2)

ist. Nun ist

und

,SchlieBlich folgt aus ( ~ u tGF(~~)(= 2 [ ~ u tGF(~) (

%.t,~~~P~~4~t Die

AnzBhl der desarnuesschen Konuruenzpartitionen

des Vektorraumes V vom RBnRe

4

S4(q3

GF(~)

- 1)(q - 1).

Sind gl, g2, g3 drei paanfeise windschiefe Geraden in einem Vektor-

raum V vom Rang 4 iiber dem K6rper K, sind ferner h,, h2, h drei 3 verschiedene Transvernalen von g,, g2, gj und iet g1 T\ hl = PI, g l n $ = P2, g 2 n h l = P,, {pl,.

.. 5 \ ,P

g2f3

$

p;3r\h3 = P5, so ist

= P4

ein h e n , dh. eine Men-

von fiinf Punkten von denen

keine vier in einer n e n e liegen. Weil GL(V) die Rahmen transitiv untereinander pemtlert, fol&,

daQ GL(v) auch auf den Tripeln

paaweise windschiefer Geraden transitiv operiert. Hieraus folgt wiedenun, daO GL(V) auf der M e n m der Regelscharen transitiv ist, da eine Reaelschar ja erade aus den s5mtlichen Transversalen von drei paaweise windschiefen Geradea besteht. Projektiv liest sich das analog, niim'lich: R;L(V)

ist ailf der Men-

transitiv. Da zu jeder Reaelschar

9 , die

der Regelscharen

ken-jugierte Re-lschar

gehort, die gerade aud den sixitlichen Transversalen von

q

besteht,

Heinz Liineburg ,der globale Stabilisator von 4 in Y X FGL(2,K) isomorph ist. Die Anzahl der Reel-

sieht man leicht, daf3 PGL(V)

PGL(V),

zu

XL(P,K)

scharen ist daher im F'alle K = GF(~) gleich

Also gilt

4242-Ei&rzf22?Z s4(s3

- l)(s2

Die Anzahl der Remlscharen in ~ ( 4 , ~ist ) ~leich

+ 1).

&Z2=giB2?4~g Die Anzahl der Reaelscharen von ~(4,~).die in einer desarnuesschen Konmuenzmrtition von ~(4.0) enthalten ist. ist 2 + 1) und die Anzahl der desarnuesschen Kon~ruenzuartigleich q(q tionen, die eine ~egebeneRe~elscharenthalten. ist ~leich~1 ( q .)1

-

Beweis. Es sei

T;-

eine desarmessche Konmenzpartition von ~(4,q).

Dann i ~ t der Stabilisator von +r in c.L(~,~)auf IT dreifach transitiv. Hieraus folgt, daO drei verschiedene Komponenten von IT stets in mnau einer oder stets in keiner Regelschar enthalten sind, deren s&ntliche Geraden zu

T

gehdren. Nun operiert SL(2,q) aher auf T(V)

in der Weise, wie in den Voraussetzungen von 4.1

beschrieben. Die

affinen Funkte der durch ~ ~ ( 2 , qbestimten ) Baerunterebenen sind dann gerade Unterrame vom Rang 2 von V = ~ ( 4 , ~ die )~ dariiberhinaus Transversalen von q halt

Ti

+

1 der Komponenten von IT sind. Folalich ent-

Repelscharen, so daO drei verschiedene Komponenten von

stets in enau einer Reaelschar lieen, die ihrerseits ganz in enthalten ist. Die 3-Teilmeneen von

IT

TI-

ir

und die in IT enthaltenen

Heinz lUnebnrg Regelscharen bilden also eine taktieohe Konfiguratlon mit v = (q2+l ), 2 k bum^ r I. wegenrr bk ist -herb q(q + I).

- Pi'),

-

-

-

Wir betrachten nun die Inzidenzstruktur der Regelscharen und der Menge DBM ist

&

(w,6i, G )

aw der Mongo

der desarguesschen Kongm611len.

@, 6 ,C ) eine taktische Konfigurationa mit den Panw-

tern v = q4(q'

- 1)(q2

und r. Wegen vr

-

+ I),b

-

bk ist daher r

-

h4(q3 1 Iq(q

-

l)(q

- I),

k

-

q(q

2

+ I)

l), q. e. d.

eine desarguessche Kongruenzpartition von ~(4,q). h a m 2 liiDt sich rr auffasaen als projektive Gerade iiber GF(~ ). Die G ~ p p e 2 XL(2,q )operiert auf lT scharf dreifach transitiv. Bildet man dee

Es sei

Ti

Tensorprodukt ~(2.q) @ GF(q) CF(q2),

so erhilt man cine Einbet-

der projektiven Gerade iiber GF(~) in die projektive GeGeFe iiber GF(q2),

so dal3 XL(~,~) auf der eingebetteten Geraden ~ ~ ( 2 ,in ~)

der richtigen Weise operiert. Nun enthalt FGL(2,q) Cruppe U der Ordnung q

+

eine zyklische

1 und diese liegt w;ederum in einer ayk-

lischen Cruppe der O r d n u n g q2

- 1 von XL(2,q2).

Da die letztere

zwei Fixpunkte hat, hat auch U zwei Fixpunkte X und Y, die jedoch beide nicht in ~ ~ ( 2 ,entha1Iten ~) sind. Ga XL(2,q2)

scharf dreifach

transitiv operiert und die Transvektionen nur einen Fixpunkt haben, folgt, daE X~(2,q)~, falls A ein Punkt von V0(2,q) Punkten von

TT

ist I X"L(2*a)1

, die nicht ?,q(q

ist, auf den

in ~ ~ ( 2 . q )lieuen, reelar ist. Daher

d l ) , SO d a ~~ F G L ( ~ , ~ ) 5 ~qI +

U 5 FGL(P,~)~ist daher U

-

I ist. we-

X~(2.q)~. Nun ist P S L ( ~ , ~C) U~, 1st

q gerade, so ist ~ ~ ( 2 , q ) XL(2,a).

1st q unmrade, so enthalt

FSL(2,q) keine zyklische Untergruppe der Ordnunr~q

+ 1.

Weil

Heinz Liineburg FSL(2,q)X

somit eine echte Untergruppe von U ist, ist ihre-0 1(q+ 1). k e r ist ( x ~ ~ ( 1~3*q(~ h6chsteni gleich z q ) I),SO dal3

-

in jedem Falle auch p~L(2.q)

nicht in v0(2,q)

auf der Menge der Punkte von

TT

, die

liegen, transitiv operiert. Diese Bemerlcung benut-

Zen wir beim Beweise von

6. Hilfssatz. Sind IT 4zil============

.Ti-'

tionen von ~ ( 4 , und ~ ) ist

zwei desarnuessche Konuruenzwrti-

9 eine Rewlschar. die in

enthalten ist. ist ferner \ -rr n r1I ) q

+

I

.T;

'md in a '

, SO ist TT =

3-

'.

Beweis. Es sei rr' die rm n konjugierte Regelsohar. Ferner sei G die Gruppe, die TTT' elementweise festlwt. Dann ist G Hilfssatz 4.1

~~(2.q).

liefert die Bistenz zweier Untergruppen H und H, von

G, die beide zurEL(2.q)

isomorph sind und die 7i bzw.

TF

'

invariant

lassen. Weil ~~(2.q)nur eine zur~(2,~)isomorphe U n t e r m p e enthalt, folgt H = HI. 1st Hun X 6 A n-tr' und X nach unserer Vorbemerkung =

lxH 1

q(q

- 1).

4 4

, so folgt

Folglich ist

rT =

9 u xH=

IT',q. e. d.

4.1p=S.4q (Prohaska). Es sei p eine Primzahl und A kine Trenslations-

.

2 ebene der O r d n u n g p Ist

eine von Elationen e n e w e Kollinea-

tionswppe von A, die nzr ~~(2,q) isomorph ist, so ist A desarnuessch.

Beweis. Weil A eine Translationsebene ist, konnen wir o. B. d. A. asmehmen, daB

einen Firpunkt hat. P sei dieser Fixpunkt. Ferner

folgt, weil A eine Translationeebene ist, daR alle Elationen von A, die von 1 verschieden sind, die

Ordnung p

haben. 1st nun

eine

p-Sylowgruppe von

x,

eine Fixgerade, da p2

so hat

der Geraden durch P ist. Weil

+1

die Anzahl

von Elationen erzeugt wird und nicht

triviale Elationen die O r d n u n g p haben, folgt, da0 nen besteht, da alle Elemente der Ordnung p in

nur aus Elatio-

konjugiert sind.

mtte nun eine weitere p-Sylowgruppe die gleiche Firgerade vie

n,

d-

so bestiinde

nur aus Elationen, da

von irgendpi seiner pESydann eine

lowgmppen er2euR.t wird. Dies kann abcr nicht sein, da

p-Gruppe wae. Also haben verschiedens pSglowgruppen verschiedene Fixeraden. Weil die Anzahl der p-S~lowqruppenvon ist, erfullen

t, der Punkt P und die Men@

I7

1gleich p + 1

der Schnittpunkte der

Fixgeraden der p-Sylowqruppen mit der uneigentlichen Geraden die Voraussetzungen von Hilfssatz 4.1. 1st daher Q ein Punkt mit P

g , so ist ~ P ' Ic, Q

aowie F Q gd~

~

L

$ T,

die F'unktrenge einer

Baerunterebene von A. Von diesen Baerunterebenen gibt es p T1,...,Tpcl

Q

+1

Stiick.

S-ien die Stabilisatoren dieser Unterebenen in der Trans-

lationspppe T von A. Diese Stabilisatoren haben alle die O r d n u n g 2

p

.

4

Ferner hat T die O r d n u n g p

, so daB T ein Vekborraum vom

Rang 4

uber GF(~) ist. Die Ti sind dann Unterraume vom Rang 2 von T. Ferner folgt, daB die Ti gerade die shtlichen Transversalen der T(x,~,) mit X 6

2

sind. Folglich bilden die T(X,~* ) eine Regelschar. Die-

se werde rnit

?

sei nun U C ?i

bezeichnet. Ferner sei \

...,

9 und es seien ri-1,

chen uesarquesschen Regelscharen, die d m

rri I,

n

Tij = Q

fiir i

11 j.

9

- -1

) {Y I ge

T(Y,~,

Kiq(q

-

-

q. e. d.

.

Es

die siimtli-

enthalten. Nach 4.6 ist

Nach 4.2 gibt es daher ein i mit

U € 'Ti i. Nach der bemerkung vor 4.6 ist daher iTi ,(? = U und folglich iT

I

't

q- .q

22,4k222ke

Netst N

- 'P.

,%,I)

auf der selben Punktmenge

und N1 =

seien Netze

Das Netx N heiBt ersetzbar durch dam

Nets N1, falls zwei verschiedene Plmkte aus bindungsgerade in

m, fhl,l')

C(2 genau dann-qine Ver-

haben, wenn sie eine Verbindungsgerade in &*

haben.

1st das Nets N =

a,1) und ist N ersetzbar durch daa Netz N 1 auch A' =

-

(7,R,IO)UnterstNktur der affinen Ebene A (9,

@, sa,1*) mit

q1= (9

\

-

(72 ,pll,l*), so ist

) u.naund '1 = (1 \

IO)UI~

eine affine Wene. In diesem Falle schreiben wir A* = A(N/NI),

1st N =

(%,'&,I)

durch N 1 =

(?;?,$l,~*)

ersetzbar und ist g

@*,

so sei

Dabei stehen die Piinktchen fiir das c&esische

Produkt der Punkt-

mit der Geradenmenge von ~ ( ). g N heiBt ableitbar, falls N(g) fiir eine affine n e n e ist. Einfaches !&haten zeigt, da13

alle g C Q

dann auch N1(h), was analog zu ~ ( g )definiert wird, fiir alle Ge-

raden h von N eine affine Ebene ist. 1st N in A eingebettet und ist N durch N f ableltbar, so heiBt auch A ableitbar und A(N/N')

heiBt

die abmleitete Ebene. 1st n die Ordnung von A, so ist n auch die

Ordnung von N und N1. Hieraus folgt, daO n die Anzahl der Punkte in ~ ( g )ist

.

Folglich ist n = m2 und N(R) ist eine Paerunterebene

von A. Aus Hilfssatz 3.2

folgt, wie Ostrom bemerkte, daB A(N/N*)

durch A und N bis auf Isomorphic eindeutig bestimmt ist. Um zu sehen, daB sich 3.2,Nwenden laBt, muO man sich nur iiberlegen, daE

Heinz Lineburg ~ ( g )und ~ ( h )fiir alle g,h &

W 'die gleichen uneigentlichen Punkte

besitzm: Um dies rm zeigen, sei rmnlichst 18(g) n I1(h) ~'(g) eine affine mene ist, gibt es m2

+m

- @. Weil

Ceraden von N, die Ce-

raden van ~ ( g )sind. Haben zwei dieser Geraden einen Schnittpankt, so ist dies ein Punkt von ~(g). Also tragen diese Geraden ins* samt (m2

+ m)(m2

-

PI) = m4

- m2

n2

- m2 Punkte von N, die nicht

rm g gehtren. Dies sind aber alle M e , die nicht auf g liegen. Es gibt also eine Cerade a von N, die sowohllPunkte von 8 als auch Weil B durch N 1 ersetzbar iat, gibt es & h e r

Punkte von h . * t

eine Cerade 1 von Nl, die mit g und auch mit h einen PunLt gemeinsam hat. Wir k6nnen daher o. B. d. A. annehmen, daS g und h einen

Punkt gemeinsam haben. Es sei P ein solcher Punlct. Die m

+1

Gera-

den von N durch P aind dann sowohl Geraden m n ~ ( g )als auch von ~(h), so da8 die m

+1

uneigentlichen M

e Ton ~ ( g )mit denen

von ~ ( h )iibereinstimmen.

?ispiel.

Es sei

9

eine Regelschar in V

'= ~(4.q)

+

und

q-sei die

rm Q konjugierte Regelschar. 1st dann

so ist N durch N 1 ableitbar. Weil

9 , wie

wir wissen, in einer de-

sarguesschen Konpenzpartition enthalten ist, fol&, 2

daS die desar-

gueasche Ebene A der O r d n u n g q ableitbar ist. A(N/N') ist die 2 der Ordnung q

.

w-

Wie wir echon bemerkta, ist &ie Ordrmng n eines durch HI ableitbaren Netzes N Quadrat einer ktiirlichen Zahl rn.

Ferner i s t r

-

m

+ 1.

Weiterhin g i l t

j) 1st g eine Gerade von N1 und ~ % l g 80 , gibt e s genau eine Cerade

h von ~ ( g m ) it P I h.

Dies haben w i r ebenfalle schon bewiesen, a l s w i r zelglen; de$ 3.2 beim Beweiae der Eindeutigkeit von A(N/N*)anwendbar i s t .

i j ) g und h eeien Geraden von N1, die keinen F'unkt meinsam haben. Es gibt dun genau eine Parallelenschar

8 von N m i t den xigemchaf-

ten: (a) 1st a eine Gerade van N, d i e Gerade von ~ ( g und ) ~ ( h i) s t , so ist a

se

(b) 1 s t a

G

g , so i a t

a mnau dann eine Gerade von ~ ( g ) ,wenn a

Gerade van ~ ( h )ist.

Beweis..Sind a,b zwei verachiedene Geraden, die beide Geraden von ~ ( g und ) ~ ( h )sind, so ist a n b =

0,

da ein Schnittpunkt von a m i t

b Funkt von ~ ( g und ) Punkt von N(h) w&e. solches

8

.

Also gibt a s hitchsten8 ein

Es s e i mm P 1' g. Nach j ) gibt ee eine Gerade a durch

P, welche Gerade von ~ ( h )ist. 1 s t lenschar, so hat

$

die durch a bestinmrte Paralle-

8 offenbar a l l e gewiinschten Eigenschaften.

W i r fiihren noch eine weitere Bezeichnung ein. 1 s t g eine Gerade von N und h eine Gerade von N1, so eei

Heine Liineburg

Entsprechend werde N1(g,h) definiert. Offenbar ist N(h,g) die ' h i denzstruktur, die aus ~ ( h )entsteht, in dem man die m g parallelen Geraden von ~ ( h )aus ~ ( h )entfernt. Dies beweist den ersten Teil des folgenden Hilfssatzes

ist ein Nets der Ordnung m und des Geredes m. 1st der

iij) N(h,g)

hrchschnitt von I1(h) rnit ~ ( g )leer, so iat ~ ( h , ~dual ) mr N1(g,h).

Beweis. Es bleibt nur zu zeigen, daB N(h,g) und N1(g,h) dual meinander sind. Es sei P E I1(h). Nach Vorsussetzung ist dann P 4 I (n). Nach j) gibt es daher eine eindeutig bestimte' Gerade P S von N 1 rnit P P P,' ll(ps)n

die C e d e von N1(g) ist. Wegen P I' pS und weil ~ ( g )#

@ ist, ist p G eine Gerade von ~'(8.h).

Es h i y

eine von g verschiedene Gerade von N, die g schneidet. Es gibt dann einenund mu. einen Punkt y8 rnit ys I y und y g I g. 1st nun P I y, so d b t es, weil ja auch y 8 auf y liegt, eine a w e x e

P , S~ I' x, denn N y

8 gemein.

vird

,nl rnit

ja durch N 1 ersetzt. Nun hat x rnit g den Punkt

Folglich ist x eine Gerade von N1(g) und damit nach j) die

einzige Gerade von N1(g), die durch P geht. Also ist x = pS. Aus

P I y folat also T J I1 pS.'Weil folgt,daB die Abbildung denztreu ist, so daB

2

allma in N W d N1 symmetrisch ist,

eine Inverse besitzt, die ebenfalls inzi-

or in der Tat eine DualitLt w n ~ ( h , ~auf ) N1(g,h)

ist, q. e. d.

Im folgenden bezeichnen wir die soeben konstruierte Dualitat rnit $(h,g).

Offenbar ist &,h)

m

8(h,g) invers.

Im folgenden ben'dtigen wir drei SStitze, die wir hier o h - B w e i a angeben.

ztltsaiz (A.

~agner)Ea sei A eine endliche affine.h e n s und

eine Kollineationsfm-uuue von A. Owriert

-

r

r auf der Men~eder Gem-

den w n A transitiv, so ist A eine Translationsebene und

r entMlt

die hanslationsmupe Von A.

Fiir einen Beweis siehe WAGNER Cqloder auch L

zZ?f,Sazz

(~kornjakov& San ~oucie).

Ist

und g eine Gerade von n, ist ferner Cj? -

fIir a l e P I g, so ist

von n, &, -

m [:6 1.

eine ~rojektioeEbene eine Tranalationaebene

Translationsebene fiLr alle Gereden h

ist eine Moufawbene.

Einen Beweis dieses Satzes, der den Fall Charakteriatlk 2 und Charakteristik ungleich 2 gleichzeitig erledigt, findet der Leser in LtiNEBUTG

tcl

~ r ~ L ~ g (Artin $g ( \ desamuessch.

& ~orn).

Ist

eine endliche M o u f ~ b e n e .so ist

Ein Beweis dieses Satzes findet sich in PICKERT C)3 oder auch in LmEBURG

L&-Je

Heinz m e b u r g

>*4. Sat% (Prohsaka).

Iet h eine

Cerade w n N',

so i s t die a f f i n e

Ebene ~ ( h desarmeseoh. ) -

Beweis. W i r kijnnen o. B. d. A.

amehmen, daB m

>2

i e t , denn ee

gibt ja nur eine a f f i n e Ebene der Ordrmng 2. Dam i e t also m

+ 1 ) 4,

so daB N mindeatens v i e r Parallelenscharen hat.

x und y seien zwei Geraden von ~ ( h ) .Ee gibt dam eine Gerade u von

~ ( h ) ,die weder zu x noch zu y p a r a l l e l iat. Wlihlt man einen Punkt auf u, der nicht zu ~ ( h geh6rt, ) so findet man eine zu h parallele Gerade v i n N1, die h nicht t r i f f t . Es s e i t e F'arallelenschar von N. Dieses

8 die durch u bestimm-

8 hat dam die-Eigenschaften von

ij). Wegen m )/ 3 gibt as echlieSlich eine Maude a w n ~ ( v ) ,die

weder zu x noch zu y noch m u parallel i s t . SchlieSlich seien

g

a,b

und X I a sowie Y I b. Ee f o l g t ~ ( a n ) I1(h) =

f a l l s wiire a eine Gerade von ~ ( h und ) daher X aber

2

8.

Andern-

E I1(h). Dann

eine Gerade von ~ ( h ) ,da X auf z l i e &

Weil z d&

Gerade von ~ ( h a) l s auch Gerade von ~ ( v vtiFe, ) Are a Widerspruch. htsprechend folgt ~ ( a n ) I1(v) =

8.

&re

*

sowohl

5, : e i n

Somit sind

&(h,a) und &(a,v) definiert. Nun i e t ~ l ( a , h )die Inaidenzstnrktur, die aus N1(a) durch h t f e r n e n der zu h parallelen Geraden entsteht,

und N1(a,v) die Inzidenzstruktur, die aus N1(a) d u r ~ hEntfernem der

zu v parallelen Geraden entsteht. Weil h zu v parallel ist, f o l g t ~l(a,h)

-

N1(a,v). ~omxti a t das Produkt o = 6(h,a) 8(a,v) defi-

n i e r t und i e t e i n Isomorphismus von N(h,a) auf N(v,a), welcher offenbar x auf z abbildet. Vertauscht man nun die Rollen von v und h und ersetzt man a durch b, so erhalt man einen Isomorphismus T: von

mit !zT = y. weii a w b parallel ist, folgt

N(v,b)

auf N(h,b)

N(v,b)

= N(v,s) und ~(h,b)= ~(h,a). Somit ist rt.eine Kollinea-

tion von N(h,s),

die x auf y abbildet. Nun ist N(h,a)

bie auf eine

Parallelenschar daseelbe wie ~ ( h )und man sieht leicht, da6 sich a zr zu einer Kollineation von ~ ( h )fortsetzen W t . Weil die Kollineationspuppe von ~ ( h )also auf der Menge der Geraden von ~ ( h )tranaitiv operiert, ist ~ ( h )nach 5.1

eine Tran~lationeebene.

Wenso gilt natiirlich, da8 N1(g) ftix a l e Ceraden g von N eine Trans1,tionsebene ist. Es sei 1 irgendeine Cerade von ~(h). Ferner sei g eine zu 1 parallele Gerade von N, die rnit h keinen Punkt gemein-

sam hat. Dam ist fl(g) eine Translationsebene und N1(g,h) ist zu ~ ( h , ~dual. ) 1st nun

der pro jektive AbschluB von ~ ( h )und P der

6chnittgunkt von 1 rnit der uneigentlichen Geraden, so folgt sue der dualen Iscmorphie m n N 1(g,h) mit ~(h,g), dafl lationsebene ist. Aus 5.2 ist, q. e.

und 5.3

eine Trans-

folgt nun, da0 ~ ( h )desarguessch

a.

6. Eine Charakterisierung der Hallebenen. Wir beginnen wieder rnit

IP=======IPPI======*===-================

mebreren Hilfssatzen.

6.1.

-

Hilfssatz. A sei eine desaruessche affine E h n e der

PI=============

q

hat -

pr.

Ist

a p-Untermppe

einen Fixpunkt und ist

deren Achse eine Gerade von A gene

der Kcllineationsm~~e von A,

(n1 2

q, so enthut

eine Elation,

(~olcheElationen heil3en Scherun-

Beweis. P sei der Fixpunkt r o n m und tionen von A, die P m F lioh ist

-

lr(

durch P ist, hat

rq(q2

w

-

77 eine

r sei die Gmppe der Kollinea-

t haben.xl)ann

- 1).

Weil q

ist

r

\~(2,~).

+ 1 die Anzahl der Geraden

F M r a d e g, die mit P inzidiert. Es sei T

die Gruppe aller Scherengen mit der Achse g. Dam ist p-Gruppe, da T ein Normalteiler von tens von p, die in r auf-ht, ist pa I r

( pr,

so daE \TT

Im folgenden sei stet8 A =

R,I~) ein in A

Folg-

iist. 8

so ist

1

4

T eine

1st pa die hdchste Po-

ITTT I ein Teiler von

q2 ist. Also ist

(\

Nun

T )( 1 I\, q.e.d.

( T , ~ , Ieine ) affine Ebene und N = (T,

eingebettetes, ableitbares Netz. Ferner sei N durch

N' = &,~,~~)abgeleitet und A1 = A(N/N*) =

,Cal,l*). 1st n

die gemeinsame Ordnung von A und A', so ist, vie wir nun wisaen, 2 n q und q = pr mit einer Primzahl p. 1st q = 2, so haben A und

-

A' die Ordnung 4 und s h d folglich beide desarguessch. Wir nehmen daher im folgenden an, daE q )2

$ ~ L ~ ~ ; f ~ a t(Prohaska). z Fhuunlct 0.

q

ist.

sei eine Kollineation von A mit dem

9 v: jede Gerade von

I(O) \ I~(o)fest. so ist

7

eine Homolo~ievon A oder von At.

Beweis. Weil die Geradenmenge van N Vereinigung von pollen Parallelenklassen von A ist und weil N duroh N' abgeleitet wid, ist

q

so-

u

wohl Kollineation von A ale auch von A'.

Als Kollineation von A l&Bt

Netz N invariant und a18 Kollineation von A1 das Netz N1.

IUt

q

einen von 0 verachiedenen Punkt von A fest, so lU3t

9

in

der zu A gehijrenden projektiven Ebene ein Viereck und damit eine Unterebene punkt- und geradenveise feet. Weil die q(q von I(0)

L

Io(0) zu dieser Unterebene gehidren, iet die Ordnmplie-

ser Unterebene mindestens ~leichq(q q(q

- 1) Geraden

- 1) - 1 7

q,

- 1) - 1.

Weil q

>2

ist, fold

so daB diese Unterebene nach 3.1 die .ganze Ebene

ist. In diesem Falle ist also

q = 1,

so daB nichts mehr m beweisen

ist.

keine Homologie von A iat. Dann gibt es

Wir nehmen nun an, da!3

I~(o) mit g'l f g. Insbesondere ist dann q f 1, so daB

ein g

0 der einzige ~ixpu$.$von

mn

oE

1

I~(~) n IA(~)ist g eine & M e von ~(h). ES gibt eomit nit P f 0. Wie wir schon bemerkten,

einen Punkt P E Io(g) '7 I;(h)

P?.

ist P f

und hq f h. We-

ist. B sei h E I : ( o )

Nach 3.2 haben ~ ( h )und ~(hf)nur einen affinen W

t

gemeinsam, nhlich 0. Die Geraden von N durch 0 sind daher die einzigen Geraden, die ~ ( h )und ~(hy)@meinsam haben. W e nun P P eine ~ Gerade von N, so wgre PP

eine Gerade von ~(h)und von ~(h?), weil

P auf h und P? auf h 7 lie&. = OP

- OPT.

Daher ware 0 I PP*~und folglich

Hieraus folfte gr

-

g: ein Widerapruch. Also ist

PP'? keine Gerade von N. Hieraus fol& wiederum, daS der uneigentliche Punkt auf P P ~ein Fixuunkt von .q ist. Dies impliziert seinerseits (PP?

= Ppq. Dies hat wiederum z u r Folge, daB

q

einen von 0 verschiedenen Fixpunkt in A hat. Dieser Wiaerspruch zeigt, d d hq =

11 (0)

I

h ist fiir alle h

L

I;(o),

so da13

2

EI~(o). Nun

ist I(O)

\

Io(0) =

alle Geraden von A', die durch 0 ~ehen

einzeln invariant lat, q. e.

(1.

Heinz Liinaburg

&& HilfssatzL A sei eine desarnuessche affine Ebene der Ordnung q. Ferner sei festlut. Wird

eine Kollineationsmppe von A, die den Punkt 0

a von Elationen erzeuat und enthalt

rm ieder

Geraden durch 0 eine von 1 verschiedene a t i o n mit dieser Geraden als Achse, so ist

~ ~ ( 2 , qoder ) aber q ist nerade und

eine D i e d e r m p w der O r d r m w , 2 (q

Beweis. .Weil

+ 1).

von Elationen erzeugt wird, deren Achsen alle durch

zu einer Untergruppe der SL(~,~)iso-

Gruppe liegen, folgt, daS

+

mornh ist. Ferner folgt, daO q

1 die Anzahl der p-S;ylowpppen

C33

ist, wenn q = pr und p eine Primzahl ist. Nach DICKSON

folgt nun, da0 entweder

A

eine

SL(P,~) oder daQ q gerade und

Diederpppe der Ordnung 2(q

6.4.

ist

und alle solche slationen in einer zur sL(2,q) isomorphen

0 @hen,

von

A

+ 1)

-

ist.

Hilfssatz. Es sei $ eine Men-

IPOIIIP===P==I=

mit (

$\

= q (q

- 1).

Ferner

sei q Potenz einer Primzahl p und 7 sei eine zur PSL(P,~) isomorphe Permutationsuru~pevon

3

.

Haben alle Bahnen von

die

lei-

che Ltin~eund hat kein Element der Ordnung p ein Fixelement in

3-& 2-Teilmennen.

so uibt es eine Zerle~ung

die von

invariant =lassen wid. so da0 der Stabilisator eines X maximale Diederfqupve von a)

q E { 5,7,11,23)

b)

o

C)

q f-

t { 7,23,47

4

\

una

e rr cine

ist. es sei d m . es ist

'Zb


11,19,29,59\

%,

SSq

fiir b f2$. b

e

a xb2 A? % b

$. C-

k?

.

Dips ist eine hlnasaufaahe, die mit Hi1 fe der Unterpppenliste der

PSL(~,?) rasch ilnd ainfach zu l h e n 1st

6.k Satz (~rohaska). A, N, N1 &y

II

(8.

DICKSON, loc. tit.).

A1 haben d i e ~ l e i c h e nB e d e u w n

====I

wie bisher. Hat A eine Kollineations,cruuve, d i e N invariant l&Bt und d i e als Gruuue vom Ranq 3 auf den Punkten von A oueriert. so ist eine d e r beiden Ebenen A

A1 desarmesech und d i e andere ist eine FIall-

ebene. Bewei~. Offenbar cplten d i e Vorauseetmaen fiir A,N e n a u dam, wenn s i e fiir A u , N 1 gelten. Nach KALLAHER CqJ

eel

und LTeBLER

sind da-

h e r A und A' Translationsebenen.

Es s e i nun 0 e i n Punkt von A. Nach V o r a u s s e t m g g i b t e s d a m eine Kollineation@mppe

von A, d i e 0 f e s t l u t und d i e auDer { o \ noch

mnau zwei F'unktbahnen hat. Ee s e i

und

LI*

I;(O).

I

O f fenbar sind

a ,

= lo(0),

%'= I(O)

itberdies f o l g t , v i e man s i c h l e i c h t i i b e r l e ~ t ,daB C l , Geradenbahnen von

d i e beiden von < O

I Q (= I

C=(

+1

= (q

\

f.7 l und k$L

sind. Ferner sind

\ verschiedenen ($1

= ( ~ ~ [ u n d

+ 1)(q2 - 1 ) und

Punktbahnen von P

(El

q{q

Z.

Es ist

- 1 ) unddaher q(a

- l ) ( q 2 - 1).

daS das k l e i n s t e ~emeinsameVielfache von (a

q(q

10!O)

und %: u n t e r - Fnvariant.

- l)(q2 - 1 ) ein T e i l e r von 1 E l i s t .

+

1)(q2

1st d = (2,q

Hieraus f o l ~ t ,

- 1 ) und

- l),

so ist

Heinz Liineburg

-

dieses k l e i n s t e geneinsame Vielfache g l e i c h c 1 q ( q 2

eine Scherung von A oder von A' ent-

A l s nSchstes zeigen w i r , daB

eine p-Sylow-

hLlt. Aqenommen,dies i s t nicht d e r Fall. Dann s e i gruppe von

x.

Weil q e i n T e i l e r von

(TI. ~ e r n e r~

i b et s e i n a E

112.

ix ( ist, i ~ t e i n T e i l e r w n q

CT m i t am rnit bm

i s t . Wenso g i b t e s e i n b e ' y 8

3

a , da ja I L t l = q

b. Weil

+1

keine S c h e q

t r e u auf den Punkten w n ~ ( b ) . Weil ~ ( b )nach

e n t ~ l t ,o p e r i e r t

nach 6.1 e i n

5.4 desarmessch ist, enthalt

welches a l s n i c h t

r

t r i v i a l e Scherung auf N(b) w i k k t . Offenbar ist dann a d i e Achee dies e r Scherung. Folglich ist jeder Punkt von ~ ( a fi ) 1 8 ( b ) e i n Fixpunkt von I T , f a l l s T das fragliche Element von

?T ist. Angenommen

operiere n i c h t t r e u auf ~ ( a ) . Dann e n t h a l t

eine von 1 ver-

schiedene XBlW Peaspektivitgt rnit d e r Achse a. Diese Perspektivit a t 1liiDt eine Parallelenschar nahme ist a

hslt,

$

g

.

Weil

miissen nach AND&

8

von A pradenweise fest. Nach An-

ts keine Scherung rnit der Achse a ent-

r.13 a l l e

Kollineationen aus

auch

festlassen. Umgekehrt miissen aus dem gleichen Grunde, da j a eine Bahn von

2- ist

auch a l l e Kollineationen aus

a festlassen. Also ist

2

=

z.

8

CT

1S d i e Gerade

Insbesondere f o l g t

8= 'cf*

so daB r~ auf ~ ( b t) r i v i a l operiert. Dieser Widerspruch z e i e , daB auf ~ ( a )t r e u operiert. J3benso f o l q t , daB

&, auf

I (b) t r e u

operiert.

Es s e i

A

-

(n-tu, O( E

(TCY

xb,

induziert eine von 1 verschie-

dene Scherung i n ~(b)). Mit Hilfe von 5.4 und 6.3 ist entweder zur S L ( P , ~ ) isomorph ode= q ist aerade undJ '/

a

ist eine Diedermp-

Heinz L i i n e b q

+ 1).

pe d e r Ordnung 2(q duzinrt wird, da ja

e i n T e i l e r von

+ I), r

q = 2

Ebauf

iz\

e i n T e i l e r von 2r(q

bestinrmt ist. Dam wkre q(q

wegen r

( 2r =

zierte q

P

i s t daher

I&,(.Wgre rmn A eine Diedergruppe

so wSre

+ l),

:112 e i n

eine p-Sylowgruppe von

iq

A.

T e i l e r von 2 r , woraus

Dann ist (

, d i e w i r wieder

1'induziert

f e s t e l a s s e n wird.

d e r- L

- 1 l 2 < 2q folgte.

q d i e m l e i c h u n g q(q

u i b t e s eine Gerade von

d-'q(q-l )(q2-1)

wobei r durch

2, was w i r a u s ~ e s c h l o s s e nhaben. Also ist

Es s e i nun

Dies impli-

A SL(~,~). I = q. Ferner

a nennen, d i e von

Scherungen van A enthiilt, induziert

keine

auch eine Gruppe von Scheauf

rungen in N t (a), d i e dann d i e Achse b haben. Es f o l & , da4

\.I

b

\ t r a n s i t i v ist.

Y'

d a m in ~ ( b eine ) Gruppe van Sche-

n m i t der Achse a. Weil w i r mnommen haben, daB

m

t r e u in-

I 1 ( b ) t r e u operiert. Nun ist

- 1)2 e i n T e i l e r von (x1 ist,

Weil d"q(q2

2(a

Hierbei haben w i r bermtzt, daB

eine U n t e r p p p e vom In-

Also enthglt

dex q, so da5 q nach DICKSON, loc. c i t . eine d e r Primzahlen 3.5,

7 oder 11 ist. Weil a lowgruppe von

zb

Somit oae-riert

-

p eine Primzahl ist, i s t V a u c h eine p-Sy-

und damit wegen anf

4 ween

(sxb(

(j (

= q

= o (q

+

1 auch von

x.

- 1) (q2 - 1) regulSr.

auf Nt(a) a l s Scherung rnit d e r Achse b.

Nun o p e r i e r t

Folglich lUt rT jede zu b p a r a l l e l e Gerade c von N1(a) fest. Weil

'rr

auch a l l e zu a p a r a l l e l e n Geraden von ~ ( b f) e s t l w t , f o l d

m i t i j ) , dan

n- i n

~ ( c eine ) nicht t r i v i a l e Translation oder Sche-

m n g induziert. I m e r s t e n F a l l e 1aBt

a l l e Parallelenscharen

von N f e s t , was zur Folge h a t , da5 ry auf ~ ( b )t r i v i a l

Heinz Liineburg operiertr e i n Widempruch. I m z m i t e n Fall hatrrFixpunkte P und Q auf b bzw. c. Nach i j ) ist FQ \ ( a und daher FQ = a. Hieraus folg-t, da0 TT i n N1(a) mehr a l e q Fixpunkte h a t , woraue wiederum

folgt. Dieser l e t z t e Widerapruch z e i g t , daB

-

n- 1

doch eine Schemmg

von A oder von A' enthiilt.

W i r konnen aue Symmetrie(pSinden annehmm, daS dene Scherung von A e n t m l t . Es s e i erseugte Gruppe. Dam f o l g t , daJ3

luteFerner der

An

in 1st

A t r e u auf

A = ~ ~ ( 2 , qoder )

Ordnung 2(a ist

ist

+ 1).

o d in

A d i e von den Sche-n A jede

A ist

= (2(q

+

l),q(q

-

e n e u g t wird,

1))

-

A XXXEEl ist.

$ j a eine Eahn von

der Ordnung 2(q

$,

f o l ~ wiet

eine Diedergruppe d e r

Hieraus f o l g t , daJ3 d i e m e n van

A d i e Diedergruppe

der Pahnen von n i n

a'. Es

A von den Schenmgen aus

a l l e d i e gleiche LSnge haben, da

aue

a tf e s t

Gerade aus

I 1 ( b ) fiir a l l e b

q ist gerade und

Weil

x.

eine von 1 verschie-

+ 1 ) und

ist t d i e I&ge

,

eo ist t e i n T e i l e r von (\dl \ $ ( )

2, so daB $ u n t e r

in l a u t e r Bahnen Ber

LSnge 2 z e r f a l l t , da d i e Elemente d e r Ordnung 2 ja Sche-n

und in

@ f o l g l i c h keine Fixgeraden

t e i l e r z d e r Crdnung q fest. Weil

+ 1 von

a

=

sind

haben. Der zyklische Normal-

1SiOt daher a l l e Geraden von

Z auf (2 t r a n s i t i v o o e r i e r t ,

f o l q t nach 6.2,

daB

.$.

Z

aus Homolo~ienvon A' besteht. Der Kern d e r Translationsebene A ' enthglt a l s o eine zyklische Unter-ppe fol*,

daB d i e s e r Kern zu ~

der Ordnung a

+ 1,

woraus

2 p ) ( isomorph ~ ist. In diesem F a l l e j e t

a l s o A r desarpeasch.

Es s e i nun

A Y ~ ~ ( 2 , q ) 1st . q

= p, so folg-t aue 4-7, daB A deearguessch

Heinz LIinebnrg ist. Ee sei also q

-

pr ) p. Die von

ist isomorph rmr PSL(~,~). Weil wieder, da0 alle Bahnen von

A auf 8 indusierte Grnppe

auf

$ tranaitiv

A in $ die ~leicheL h i p haben.

gibt es wegen q ) p eine Zerlegung a von

6.4

operiert, folgt Nach

&- in 2-Teilmengen,

A invariant mlaasen wird. Fs sei Z der Zentralisator von A in r.Dann ist / Z isomorph zu einer Untergruppe von die van

rL(2.13).

da T~(2.q) die Automorphismengruppe von ~~(2.q)ist.

Hieraus folgt, da6 ist

Ix / Zit

daB q2 da

/

(

= rq(q2

-

Z I ein Teiler von rq(q2 - 1) ist. Also 1). Ferner ist

- 1 ein Teiler von

r d ( 1 I ist.

1x1

= ud-'q(q2

- 112,

Nun operiert Z auf

ad

trivial,

Z ja insbesondere alle maximalen Diedergruppen von A zentrali-

aiert. Hieraus folgt, dsS die Untergruppe den von

% festlSt, htichstens den Index 2 in

jedes Element ron Weil

A

folut,

von

A entweder eine Homoloqie von

Z, die alle Gerahat. Nach 6.1

ist

A oder eine von A'.

nicht VeraPniuunu von zwei echten Unterpppen aei kann,

(tsB

/\

entweder nur aus Homologien von A oder nur aus Homo-

logien von A' beeteht. (ES ist nicht verboten, daf3

A

Elemente ent-

halt, die Homologien von A und von A' aind. ) Nun ist dr ( q und folglich q

+ 1 < \ Z 1,

1 woraus -i(q

-1

+ 1) ('/ /\( fol~t. Dies

impli2 ziert wieder, daI3 der Kern von A oder van A' gleich GF(~ ) ist, q.e.d.

E : ~ ~ ~ & C.uAndr6, J.,

C21 And&,

ijber nicht desarguessche Fbenen mit transitiver Trans-

j.,

iiber PerspektivitLten in Wdlichen projektiven Ebenen.

Arch. Math. 6(1954), 29-32. [jl Dickson, I. E.,

Linear groups. Neudruck: New York 1958.

[47 Kal1aher.M.

On finite sffine planes of rank 3. J. of Almbra

J.,

Heinz Liinebg I3 (19691, 544-553.

(31 Liebler,

R. A.,

,

Finite affine planes of rank three are transla-

tion planes. Math. 2. 116 (1970)~89 l67 Liineburg, H.,

- 93.

Lectures on projective planes. University of

Illinois at Chicam Circle. Chicago 1969. i73 Pickert, G.,

Projektive menen. Berlin-Giittingen-Heidelberg 1955.

C83 Prohaska, O.,

M l i c h e ableitbare affine Ebenen. Geometrise

Dedicata 1 (1972). C93 Wagner, ,.A

On finite affine line transitive planee. Math. Z.

87 (1965)~1-11.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E . )

8.A . THAS

4-GONAL

Corso tenuto

CONFIGURATIONS

a Bressanone

dal

1 8 a1 2 7

Giugno

1972

JoAo THAS

4 - ~ N ACOTJFI L GURATIONS J. A. THAS

1. All known 4-gonal configurations

[4] is an inci.dence structure ~=(~,B,I),with an incidence relation,satisfying .. . the following axioms (i) each point is incident with r lines (r>,2) and two distinct points are incident.with at most one line; (ii) each line is incident with k points (k22) and two distinct lines are incident with at most' one point; (iii) if x is a point and L is a line not incident with x,then there are a unique point x' and a unique line L' such that xILtIx'IL. 1-1. A finite 4-gonal configuration

(BI

If IP( =v and =b,then v=k(kr-k-r+2) and b=r(kr-k-r+2). In [7J D.G.Higman proves that the positive integer k+r-2 divides kr(k-1) (r-1 ). :,:oreover,underthe assunption th'at k > 2 and rB2,he shows that r C (k-1)2+I and B C (r-1)2+1.

,...,

,...,

Let P = ( x . .I1 i,j=1,2 k1 and B=(L1,L2 Lk K M = J ' IS-2' Ek),ivhere k 5 2.Incidence is defined as follows : IL1B i = l , x . 1 J.IDL1wj=l.Then S=(P,B,I) is a 4-gonal configuration wltk parameters k=k,r=2,v=k2,b=2k. 1.2.

...,

1 .?. (J.Tits [4] ) Ke consider a nori-sin~ularhyperqundric Q of index 2 of the projective space ~~(d,~),with d=3,4 or 5.Then the points of Q togetker with the 1i:ies of Q (which are the subspaces of maximal dimension on &) form a 4-gonal configuration with parameters k=q+l7 r-7 ~ v=(q+1 9 l2,b=2(q+l),vihen d=3; k=r=q+l,v=b=(q+l)(q 2+l),when d=4; k=q+l,r=q2+l ,v=(q+l )(q3+l ),b=(q 2+I)(q 3+I1, when d=5.

(~.T!its [ 4 ] ) Let H be a non-singular Hermitian primal of the projective space ~ G ( d , q ) , ~ = ~ ~ ~ d=3 . ~ or f $,then the points of H together with the lines of H form a 4-gonal configuration with parameters for d=3 : k=q+l,r=l+E,v=(q+l) (l+q\lii),b=(l+\&) (I+~\T~); for d=4 : k=q+l,r=l+qK,v=Cq+l)(l+q2m ,b=(~+~\~i) (~+~~\lq). 1.4.

fig

1.5. its [4] ) Let d=2 (resp. d=3) and consider a (q+l)-arc (*) K (resp, a (q2+I)-cap (*) K) of ~ ~ ( d , q(not ) at the same time d=3 and q=2).ISet ~G(d,q) be embedded as a hyperplane H in pG(d+l,q)=p.Define points as (i) the points of p-~,(ii) the hyperplanes X of P for which Ixn~I=I,and (iii) one new symbol u.Lines are (a) the lines of P which are not contained in H and meet K (necessarily in a unique point),and (b) the points of K.Incidence is defined as follows : Points of type (i) are incident only with lines of type (a);here the incidence is that of P.A point X of type (ii) is incident with all lines C X of typ'e (a) and with precisely one line of type (b),namely the one represented by the unique point of K in X,Finally,the unique point u of type (iii) is incidcnt with no line of type (a) and all lines of type (b).~he incidence structure so defined is a 4-gonal configuration xith parameters k=r=q+l,v=b=(qt 1 ) (q2+l ) ,when d=2; k=q+l,r=q2+l,v=(q+l ) (q3+~ ) ,b=(q2+l ) (q3+1 ) when d=3. 1.6. F!EGURK. The points of ~ ~ ( 3 , ,together q) with the totally isotropic lines with respect to a symplectic polaritym form a 4-gonal configuration N(q) with parameters k=r=q+l, v=b=(q+l ](q2+l).~his configuration is the dual of the +gonal.configurztion arising from a non-singular hyperquadric of index 2 in PG(4,q) [ 2 ] .

.

,

(*I A t-arc of ~ ~ 6 2 , q(resp. ) e t-cap of PG(3,q)) is a set or t points of PG(2,q) (resp. ~~(3,q)),no three of which are collinear.

J. A. THAS

1.7. (~.~.Payne [g]) Consider a 4-gonal configuration S with parameters k=r=s+l,v=b=(s+l)(s2+1 ).If x and y are distinct points of S,let zo,zl, zs be the points collinear with both x and y.Then the pair (x,y) is called reguler provided that any point collinear with two of the zits is collinear with all of them.The point x is regular if the pair (x,y) is regular for all points y,y#x (for example, every point of the 4-gonal configuration ~ ( q )is regular). Dual definitions for regular pairs of lines and regular lines. Let x be any regular point of S,and let L1 and L2 be two lines through x.A 4-gonal configuretion S* is defined as follows.Points of S* are precisely the points of S which are not collinear in S with x.The lines of S* are of two types.Those of type (i) are the lines of S not incident with x,and those of type (ii) are indexed by pairs [xl,x2] of points such that xiILi,xifx,i=1,2.For lines of type (i) the incidence is that of S restricted to points of S*.The line [x1,x2] is to be incident (in s*) with precisely those points of S different from x which are collinear in S with both xl and x2.The incidence structure so defined is a 4-gonal configuration with parameters k=s,r=s+2,v=s3,b=s3+2s2 There follows that there exists a 4-gonal configuration with parameters k=q,r=q+2,v=q3,b=q3+2q2,where q is any prime power.

...,

.

EETLRK. For particular cases see

[ll

and [6].

1.8. (~.d.Thas [16]) In ~ ~ ( 3 n ,q),q=2h -1 and l , n ,we consider qn+2 (n-1 )-dimensional subspeces PG" ) (n-I ,q) , , n PG(~+2)(n-l ,q) ,every three of them being joined by p~(jn-l,q) (with such a set of subspaces there corresponds a (qn+2)-arc of the projective plane over the'total matrix algebra of the n x n matrices with elements in GP(~) [ 1 4 ] ) .

...

-

Let PG(3n-1,q) be embedded as a hyperplane H in ~~(3n,q)=P. Define points of the 4-gonal configuration as the points of P-H.Lines are the n-dimensional subspaces of P through pGCi)(n-1 ,q) ,i=1,2,. ,qn+2,which are not contained in H. Incidence is that of P. The incidence structure so defined is a 4-gonal configuration with parameters h k=qn,r=qn+2,v=q3n,b=q3n+2q2n (q=2 . ) l n ,

..

TIlE PARTICULAR CASE n=l. For n=l we obtain a 4-gonal configuration with parameters k=q,r=q+2,v=q3,b=q3+2q2 (q=2?, arising from a (q+2)-arc in ~ ~ ( 2 , q (so ) we find again the model constructed by ILHall,Jr. [ 6 ] ) .

1.9.

(~.~.Thas [I611 In ~G(3n-l,q),q=ph and n>l,we consi-

...,

der qn+l (n-I )-dimensional subspaces PG" ) (n-1 ,q) , n P G ( ~ )(n-1 ,q) ,every three of then being joined by pG(3n-1,q) (with such a set of subspaces there corresponds a (qn+l )-arc K of the projective plane over the total rnatrix algebra of the n x n matrices with elements in GF(q) 4 we have proved that through PG(~)(n-1, q) , [14] ) .In i=1,2,. ,qn+l ,there passes one and only one subspsce ~~(~)(2n-l,q)of PG(3n-1 , q ) which has no point in common with the set PG("(~-1 ,q)~.. .uPG(~-' '(n-1 ,q)~P~(~+~)(n-l, n q) .UPG(~ )(n-1 ,q) (with the qn+l spaces ~~(~'(2n-1 ,q) there correspond the qn+l tangent lines of the (qn+I )-arc

-

..

u..

K

+'

04).

Let ~~(jn-1,q) be enbedded as a hyperplane H in ~~(3n,q)= =P.Define points of the 4-gonal configuration as (i) the points of P-H (ii) the 2n-dimensional subspaces X of P for i;;hich x~H=PG(~)(~~-I ,q) ,i€ \1 ,2,. ,qn+l) (iii) one new symbol u.Lines of the configuration are ( a ) the n-dimensional subspaces of P which are not contained in H and pass through one of the spaces PG" )(n-1 , q ) ,pG(*)(n-l ,q) , n P G ( ~'"(n-1 ,q),and (b) the spaces PG(' )(n-1 , q ) ,. ,

..

,...

..

n

) (n-1 ,q). Incidence is defined as follows : Points of type (i) are incident only with lines of type (a) ; here the incidence is that of P.A point X of type (ii) is incident with all lines C X of type (a) and with precisely one line of t pe (b),namely the one represented by the unique space PG(if(n-l ,q) in X.Pinally,the unique point u of type (iii) is incident with no line of type (a) and all lines of type (b). The incidence structure so defined is a 4-gonal configuration with parameters k=r=qn+l,v=b=(qn+l ) (q2"+1 ) (q=ph and n ).

a1

THE PARTICULAR CASE n=l. For n=l we obtain a 4-gonal configuration with parameters k=r=q+l,v=b=(q+l ) (q2+1 ) (q=ph 1, arising from a (q+l)-arc in PG(2,q) (so we find again one of the models constructed by its). 1.10. (J.~.~has [16]) In PG(4n-l,q),q=ph and nJ1 (and not at the same time q=2,n=l),we consider q2n+l (n-1 )-dimensio2n ) (n-1 ,q) (satisfying nal subspaces PG(' ) (n-1 ,q) , ,PG(~ the following (i) every three of these spaces are joined by a (3n-1)dimensional subspace of ~ ~ ( 4 n - 1q) ,; (ii) every four of these spaces are contained in a ~~(3n-1,q) or are joined by PG(4n-1,q); (with such a set of subspaces there corresponds an ovaloid K of the threedimensional projective space over the total .natrix algebra of the n x n matrices with elements in GF(~) [14] ).In [la we have proved that through PG(~)(~-I ,q), i=1,2,. ,q2n+l,there passes one and only one subspace PG(~)(~~-I, q) of ~ ~ ( 4 n -q) 1 ,which has no point in common with the set PG("(n-1 ,q> ~~(~-l)(n-l ,q)~F'~(i+' )(n-1, 2n q)V.. .V PG(P +')(n-1 ,q) (with the q2n+l spaces PG(~)(J~-1, q) there correspond the q2ntl tangent planes of the jq2n+l )-cap K 11411. Let PG(4n-1,q) be em:edded as a hyperplane H in ~~(4n,q)=

...

-

..

u..."

+'

=P,Def ine points of the 4-gonal configuration as (i) the points of P-It Cii) the 3x1-dimensional subspaces X of P for which xnH=~~(.~'(3n-l ,q) ,iE[1,2,. ,q2%l\ Ciiil one new symbol u,Lines of the configuration are (a) the n-dimensional subspaces of P which are not contained in H and pass through one of the spaces PG(' )(n-1, q) ,PQ(2)(n-1 q),,,,, 231 PG(Q '''(n-l,q),and (b) the spaces ~~(l)(n~),~),

..

,

...,

P G ( Q ~ ~)+(n-1, ~ q);Incidence is defined as follows : Points of type (i) are incident only with lines of type (a) ; here the incidence is that of P.A point X of type (ii) is incident with all lines C X of type (a) and with precisely one line of ty e (b),namely the one represented by the unique space PD(iP(n-l ,q) in X.Pinally,the unique point u of type (iii) is incident with no line of type (a) and all lines of type (b). !The configuration S so defined is a 41gonal configuration with parameters k=$+l ,r=qZn+1 ,v= (qn+l ) (q3n+1.),b=(q2n+~) (q3n+l (q=ph ,n,l,and not at the same time q=2,n=1). THE PARTICULAR-CASE n=l. For n=l we obtain a 4-gonal configuration with parameters kq+l ,r=q2+l ,v=(qtl.) (q3+I 1, b=(q2+l ) ( q3+1 ) (q=ph and q> 2) ,arising from a (q2+I )-cap in ~ ~ ( 3 , q(so ) we find again one of the models constructed by J.Tits). 1.11. OPEN QUESTIO??. Do the general constructions 1.8.,1.9., 1.10. yield configurations which are not isomorphic to configurations provided by the particular cases n=l ? 2,

2.1.

4-gonal configurations of order n Definitions

A 4-gonal configuration of order n is a 4-gonal configuration with parameters r=k=n+l,v=b=(n+l)(n2 +I). Tlet s=(P,B,I) be a 4-;or.;l confi~urntionof order n and

J.A. THAS

let x and y be distinct points of S.The trace of x andy is defined to be the set trlx,y)=[all ZEPU z is collinear with both x and y1.H (x,y),x,y€P,is a regular pair of S and if z, and z2 are distinct points of tr[x,y],then tr{zl,z2) is said to be the span of x and y (notation : sI?{x,y) 1. 2.2. Theorem (~.~.~ingleton.~.~.~enson,~.~.~a~ne) Let x be a regular point of a 4-gonal configuration S of order n.Let Tt, be the incidence structure whose points are the points of S collinear with x and whose lines are the spans of the (neces~arilyregular) pairs of distinct points of TC,.Then with the natural incidence relation is a projective plane of order n.

>

2.3.

Theorem (J.~.~hasl

Consider a 4-gonal configuration S=(P,B,I) of order n (n)2),and suppose that K is a pointset of S satisfying the following (i) I K I = ~ ~;+ I (ii) every line of S is'incident with one and only one point of K (i.e. no two points of K are collinear) ; (iii) for any three distinct points x,y,z of K there is some point w collinear with ali of x,y,z. Then n=2h and s=(P,B,I) is the 4-gonal configuration ~ ( n ) arising from a symplectic polarity Tt of PG(3,n) (see 1.6.). Moreover K is a (n2+l)-cap of ~~(3,n),forwhich the tangent lines coincide xith the totally isotropic lines of the polarity n.

-

Pro~f. First of all we define the following incidence structure I*(K)=(K,B*,I*): the points of I*(K) are the elements of K ; the blocks of I*(K) (i.e. the elements of B*) are called circles and are defined as follows : the n+l lines incident with an arbitrary point PEP-K are incident wi.th n+l points

of K.The set defined by these n+l.,pointsis a circle C of I*(K) ; if x E K and CEB*,then xI*C if and only if xEC. Now we show that I*CK) is an inversive plane of order n. a) The number of circles is at most ~ ~ - ~ ~ = n ~ + n (iii) .~rom there follows that through every three points of K there Dasses at least one circle.So the number of circles l a at

*

least = n3+n,Consequently i~*l=n~+n.~ence it appears that through every three points of K there passes 6ne and only one circle. From the preceding there also follows that every circle C is defined by one and just one point PEP-K.le say that p is the nucleus of C and that the lines incident with p are the tangent lines of C, b) Consider a circle C,a point x e C and a point yEK-C. Vie shall prove that there is a unique circle C1,with y e C 1 and CAC'={X]. First of all vre remark that the number of circles,different from C,through x equals !{all PEP-K113 L E B with x1~1p>(-l = n(n+l)-1 = n2+n-1,Next we determine the number of circles which pass through the points x and zeC,where z#x.This number equals ){all p fP-K 11 3 L,L' C B with XILI~IL*IZ~\ = n+l (to obtain this result,take an arbitrary line L incident with x;from axiom (iii) in the definition of a 4-gonal configuration it follows that there is a unique point PEP-K and a unique line L' such that zIL'IpIL; so the number we are looking for equals the number of lines L incident with X),NOW it is easy to deduce that there are exactly n-1 circles C1,with cnCt=ix\. Next we consider the nucleus p of C,and the tangent line L of C which is incident with x.If qIL and'q${x,p) ,then we prove that the circle C1 with nucleus q is tangent to C. Suppose that CC\C1={x,z) and that L 1 (resp. Ln) is the tangent line of C (resp, C') at the point z.Then zIL'IpIL and

J.A. THAS

zUWIqIL,in contradiction with axiom (iii)- in I.1.So there results that Cnct=[x\.In this way we obtain the n-1 circles C*,with CAC*=jx).Finally,we consider the unique point q and the unique line &defined by yIMIqIL (qg ix,pu/~,ICI>O for all CEB*,and there exist a x E K and a CEB* with X ~ C * From (a),(b) and (c) there follows immediately that I*(K) is an inversive plane of order n.Next we prove that n is even. For that purpose we consider a circle C,a point x E C and a point y6K-C.Through y there passes a unique circle C*, with C A C*=ix).No.# we take an arbitrary point z C,with z#x,and we consider the unique circle Cv,with z E C n and C * ncw={g) .we shall prove that (CnCw(=2.Suppose a moment that C ~ C ~ ~ = {.The Z ) nucleus of C (resp. C*,cW) is denoted by p (resp. p*,ptt),and the tangent line of C* or Cw (resp. Cw or C,C or cl) at the point y (resp. z,x) is denoted by L (resp. L* then p#L,pILWIp*IL and pILfIpwIL,incontradiction with axiom (iii) in 1.1.Consequently \CnCn\=2(we also remark that X+CAC~).I~ follows immediately that ICI-l= =n is even,and so I*(K) is an egglike inversive plane of even order n [3,q .There results that n is of the form 2h Since I*(K) is an egglike inversive plane of order n=2i there exist a (n2+1)-cap K t in PG(3,n) and a bijection Q of K1 onto K,such that for every plane H of ~ ~ ( 3 , nwith ) I!lnK*I > 1, ( K * A H ) ~is a circle of I*(K).If w(n)=(P1 ,B* ,E) j s the 4-gonal configuration arising from the symplectic polarity Tt defined by Kr,then we define as follows the bijection Y of P I U B * onto P U B : (i) x V = xC ,when x E K * ; (Ti) let xEPt-Kt and call H the polar plane of x with respect to the symplectic polarity D;then xQ is the nuclee us of tT1e circle (K*~H)' of I*(M) ;

(iii] let L E Bf.and suppose that 'L contain8 the ,point x of Kt ;if H is a plane with L C H and IHAK*l>t, then 'L is the tangent line of (H~ K )' I at the .point xY

.

From the preceding there follows immediately that Y is a bijection of P U B ' Qnto P U B and that .L€x xxYI L ' , for all xEP' and LEB'.Consequently Y is an ieomorphism of ~ ( 2 ~onto ) s=(P,B,I) ,with''K =K.And so our theorem is completely proved. Corollaries a) If.q is an odd prime power then the maximal number of points of ~(q),no two of which are collinear,is less than l+q

.

2

b) Consider a 4-gonal configuration s=(P,B,I) of order n (n>2),and suppose that K is a pointset of S satisfying the following (i) I K I = ~ ~;+ I (ii) no two points of K are collinear ; (iii) every point of K is regular. Then n=2h and s=(P,B,I) is the 4-gonal configuration . Y(n) arising from a symplectic polarity'n of ~~(3,n).~oreover K is a (n2+1)-cap of ~~(3,n),forwhich the tangent lines coincide with the totally isotropic lines of thg polayity Tt

.

-

Proof. It is sufficient to prove that for any three distinct points x,y,z of K there is some point w of S collinear with all of x,y,z.So we consider three distinct points x,y,z of K.Then tr{x,y\ and tr{x,z) are two lines of tke projective plane lTx defined by x (see 2.2.).The common point w of these two lines of TT, evidently is collinear with all of x,y,z. 2.4.

Theorem (Z.~.Thas)

Suppose that the 4-gonal configuration S=(P,B,I) 'of order n has a regular point x and a replar pair of lines

(Y;z) ,where Y and Z are not concurrent.If no line of- tr$,z) is incident with ,,then n'is even.

-

Proof. Consider the get ~=lallp €P 113~ €tr[~,~l with ~IL) =\all p C P 1 1 3 L ~s p i ~ , ~with ) PII,] (we remark that x#~).The set Q contains exactly n+l points of the projective plane nxdefined by the regular point x (each of these points is incident with an element of tr {Y,z] (resp. sp{Y ,z)) ;conversely,each line of tr\~,z](resp. s~\Y,z)) is incident with one of these n+l p6ints) .As x 4Q the set K of these n+l points does not contain ,.Taking account of the regularity of the pair (Y,Z) it,is easy to show that each line of m, through x contains exactly one point of the set K (if a line of nxthrough x would contain at least two points of K,then there follows immediately that XEQ). Next we shall prove that K is a (n+l)-arc of the projective plane =,.Suppose that the points u,v,w of K are col'linear in the plane =,.Let U be the line of triy,~) which is incident with u,and let V (resp. '!/) be the line of sply,z) which-is incident with v (resp. w).If a and b are the points defined by UIaIV and UIbIW,then evidently afb.As u,v,w are collinear in Rx,we have b €tr{u,v). There follo~~s immediately that a=b,a contradiction.So we conclude that K is a (n+l)-arc of the projective plane I T , . From the preceding there follows immediately that the tangents of the (n+l)-arc K all meet in x.Pence it appears that n is even, Corollaries a) If the 4-gonsl configuration s = ( P , E , I ) of order n has a regular point x and a regular line Y,with xlY,then n is even. Proof. Let x* and Y* be defined by xIY*Ix*IY.If Z is a line which is not concurrent with Y or Y1,then (Y,z) is a regular pair of lines.Fv~oreoverY and Z are not concurrent and no line of tri?',~] is jncide~twith r;.F'roa the theorem

-

J.A. THAS

there follows immediately that n is even. b) If the 4-gonal configuration S-(P,B,I) of order n,n odd,contains two regular points,then S is not self-dual. Proof. Let x and y be two regular points of S and suppose that g is an anti-automorphism of S.Then at least one of the reeular lines x6,yg is not incident with at least one of the regular points x and y.From (a) there follows immediately that n is even,a contradiction. c) The 4-gonal configuration ~ ( q ,q=ph ) and p odd, is never self-dual (see also [2] and D5]). Proof. Every point of ~ ( q )is regular,and so from (b) there follows immediately that W(q),q odd,is never self-dua1

-

-

.

REFERENCES 1. R.W.AHRENS and G.SZEKERES,On a combinatorial generalization of 27 lines associated with a cubic surface,J. Austral, Math. Soc.,X (1969),485-492. 2. C.T.BENSOH,On the 'structure of generalized quadrangles, J. Algebra,l5 (1 970) ,443-454. 3. P.DEN;BO:';SKI and D.R.HUGHES,On finite inversive planes, J. London Kath. Soc.,40 ('1965),171-182. 4. P.DEM301SKI,wFinite geometries," Springer-Verlag,l968, 375 PP. 5. W.FEIT and D.G.HIGNiN,The nonexistence of certain gencralized polygons,J. Algebra,l (1964),114-131. 6. K.HALL,Jr.,Affine generalized quadrilaterals,Studies in Pure Mathematics,ed. by L.lLirsky,Academic Press,l971, 113-116. 7. D.G.HIGVAN,Partial geometries,generalized quadrangles and strongly regular graphs,Atti del convegno di geometria combinatoria e sua applicazioni,Perugia (1971). 8. S.E.PAYlE,Affine representations of generalized quadran-

J.A. THAS

gles,J. Algebra,l6 (1 970),473-485. 9. S.E.PAYNE,Nonisomorphie generalized quadrangles,J. Algebra,l8 (1971),201-212. lO.S.E.PAYNE,The equivalence of certain generalized quadrangles,J. Comb. Theory,10 (1 971) ,284-289. ll.B.SEGRE,Introduction to Galois geometries,Atti della Accad. Raz. Lincei,Mem. cI. di Sc. Fis. Illat, e Nat.,8, Sez. I",F~SC. 5 (19671,137-236, 12.R.R.SINGLETON,~~inimal regular graphs of maximal even girth,J. Comb. Theory'l (1 966),306-332* 13.J.A0THAS,Een studie betreffende de projectieve rechte matrices over de totale matrix algebra M (K) der 3 x 3 3 met elementen in een algebraIsch afgesloten veld K,Verh. Kon. V1. Acad. Bet.,Lett. Sch. K. van Belgi&,Kl. der Wet, 31,no. 112 (19691,151 pp. 14.cT.A.THAS,The m-dimensional projective space s,(%(GF(~))) over the total matrix algebra %(GF(~)) of the n x n matrices with ele~llentsin the Galois field GP(q),~end. di :Bat., (6) 4 (1971 1,459-532. 15,J.A.TiS,Ovoidal translation planes,Arch. der Math., (23) 1 (1972),110-112. 16.J.A.THAS,On 4-gone1 configurations,Geometriae Dedicata (to appear). l7.J.TITS,Sur la trialit6 et certains groupes qui sten d6duisentl~ubl. Llath. I.H.E.S. Paris,2 (1959),14-60. 18. J.TITS,G6om6tries polyedriques et groupes ~ i m ~ l e s , 2 ~ Reunion ;:ath. d'expression latine (Firenze-~ologna1961), Cremonese,Roma,66-88. 19.J.TITS,G6om6tries 2oly6driques finies,Rend. di Wat.,23 (1964),156-165.

-

-

CENTRO INTERNAZIONALE MATEMATIC0 ESTIVO (C.I. M. E . )

H.P. YOUNG

AFFINE TRIPLE SYSTEMS

C o r s o tenuto a B r e s s a n o n e

dal

1 8 a1 2 7 G i u g n o

1972

AFFINE TRIPLE &STEMS

H. P. Young

1. Matroids I n t h i s paper we present a new c l a s s of f i n i t e geometries (matroids) t h a t have t h e same s t r u c t u r a l r e g u l a r i t y found i n t h e c l a s s i c a l a f f i n e and p r o j e c t i v e geometries, and whose c o S r d i n a t i z a t i o n by a c l a s s of Moufang loops provides i l l u m i n a t i n g l i n k s between geometric and

A more d e t a i l e d account of t h i s work may be found

a l g e b r a i c concepts, i n [91.

A matroid empty s e t

dl

is a f i n i t e s e t

(E,@)

M

of subsets of

e n t s e t i s independent; and (12)

A

A,

together w i t h a non-

E, c a l l e d independent s e t s , and s a t i s f y i n g

t h e following axioms of independence.

independent s u b s e t s of

E

(11)

For every

Every subset of an independ-

A = E, every two maximal

have t h e same cardinality, c a l l e d t h e

rank of

r(A). For any

A 5 E, t h e submatroid of

t h e matroid on s e t

A

with independent s e t s

A k - f l a t i s a maximal s e t having rank k.

(r(E)-1)-flats. only i f :

(Hl)

member of

kL

in

H1,

H2

of

w.

M

on

A, denoted by

dA = {J

E

4:J

WA, is

5 A)

.

The hyperplanes a r e t h e

';CI is t h e hyperplane family of a matroid on E i f and The members of

a r e proper subsets of

is a proper subset of another; and

and

x

E

E, (HI n Hz)

(H2)

E

and no

For any d i s t i n c t

u {x) i s contained i n some member

A matroid design is a matroid i n which a l l t h e hyperplanes have

the same c a r d i n a l i t y . t h a t f o r every

A p e r f e c t matroid design (PMD) i s a matroid such

k, a l l k-flats have t h e same c a r d i n a l i t y , a(k).

[See [7]

H. P. Young

and [81) A S t e i n e r t r i p l e system

S = ( ~ , k ). w i t h

a rank 3 PMD, whose hyperplane family We w i l l c a l l t h e 1 - f l a t s of

A subset

t h e unique l i n e cont'aining of

A,

c12(A)

E, c a l l e d

x

and

bases of

y

S

system.

of planes of

A.

x,y a A

The 2-closure

is t h e i n t e r s e c t i o n of a l l 2-closed s e t s containing

is an (order 3) a f f i n e plane, then we c a l l

If

S, a r e j u s t t h e non-

is contained i n

The 2-closure of a b a s i s w i l l be c a l l e d a plane of of

w. The

is 2-closed i f f o r every

A5 E

S = (E,*)

S. S

If

an affine t r i p l e

i s a n a f f i n e t r i p l e system and

a ( 1 ) = 1, a ( 2 ) = 3,

a ( 3 ) = 9,

A.

every plane

@

is t h e family

S, i t is r e a d i l y v e r i f i e d by (Hl) and (H2) t h a t

a rank 4 PMD i n which

S.

is j u s t t h e s e t of blocks of

S p o i n t s and t h e 2 - f l a t s

maximal independent s u b s e t s of collinear triples.

d

h = 1 i s a n example of

and

is

(E,@)

a(4)

I E ~.

a

I n [ 3 ] , M. H a l l c o n s t r u c t s a n a f f i n e t r i p l e system whose p o i n t s and l i n e s a r e n o t isomorphic t o t h e p o i n t s and l i n e s of an a f f i n e geometry of o r d e r 3; i.e.,

whose 2-closed s u b s e t s a r e n o t t h e subspaces of some AG(n,3).

The rank 4 PMD associated with any such a f f i n e t r i p l e system w i l l be c a l l e d a H a l l matroid.

I n t h i s paper we w i l l show t h a t H a l l matroids can

be c o k d i n a t i z e d by a c l a s s of Moufang loops, and by t h i s method we a r e a b l e t o c o n s t r u c t an i n f i n i t e c l a s s of them and analyze t h e i r geometric properties.

2.

Cob'rdinatization by Loops Bruck [ I ] seems t o have been t h e f i r s t t o n o t e s e v e r a l ways i n

which t r i p l e systems may be coordinatized by loops. (See a l s o [41) we develop some of these i d e a s f u r t h e r .

Here

H. P. Young

For every S t e i n e r t r i p l e system

S = ( ~ ~ $ 1we may make

a quasigroup by defining a binary operation x o x = x For Then

x

Z y,

x

-

y

0

E

into

a s follows.

"0"

(idempotence)

z i f and only i f

is a triple.

{x,y,z)

is commutative and furthermore s a t i s f i e s

"0"

x

0

(x

y) = y

0

for a l l

x,y

E

E

.

However, t h e r e a r e more u s e f u l ways of a s s o c i a t i n g quasigroups, and i n p a r t i c u l a r , loops, with a given t r i p l e system

Let

S = (E,&).

always denote t h e binary operator defined above on t h e p o i n t s of t h e

"0"

t r i p l e system.

For a given

e

E, d e f i n e t h e loop of

S

based a t

e,

G (S), a s follows.

For a l l

(1)

x,y

E

E,

xy = ( e

X)

0

(e

0

It then follows from t h e p r o p e r t i e s of

(i)

xe = ex = x

(ii) (e

0

Thus

xx = e x)

0

x

x = x

Ge(S)

for a l l

x

E

E

(e

0

x) = e;

"0"

y)

.

that:

( e is t h e i d e n t i t y of

i s t h e unique i n v e r s e of Q

o

(iii)

Ge(S);

x, s i n c e

xy = yx

for a l l

is a commutative loop of exponent 3.

xy = e

0

(x

Y

-1

xy

y.

S.

For example,

y).

I n view of ( i i ) above we s h a l l sometimes w r i t e Thus

and

Note t h a t t h e r e is

nothing sacred about t h i s way of a s s o c i a t i n g a loop with we could j u s t a s w e l l have defined

x

x

-1

is t h e t h i r d point of t h e unique t r i p l e containing

for

x

x-'

2

. and

O r

-1 -1 xy=x o y

(2)

.

Now can we, given a conimutative loop identity

G

of exponent 3 with

e , r e v e r s e t h e process and construct a t r i p l e system whose

H. P. Young

loop based a t

e .is

G?

In'general, we cannot.

containing d i s t i n c t elements

, and

{x-',y-',q~~

x

-1

'

and

y

-1

Indeed, t h e t r i p l e .

would have t o be

s o we must a l s o have

x-l = (y-l)-l(v)-l and y-I = (xy)-l(x for a l l Since

(x-l)-l

= x

x

and

y

-1 1

1-

.

i n a commutative loop, t h e above r e l a t i o n s

a r e equivalent t o y(xy)-l

= x-I

for a l l

x,y E G

,

which i s known a s t h e weak i n v e r s e property f o r a commutative loop [61. (3)

Thus, i f

is a commutative loop of exponent 3 with t h e weak

G

d

inve;se

property and we l e t

(G,&)

is a t r i p l e system whose loop based a t

= ~ { x - ~ , ~ - ~ , x ~E ) G, : x x, ~# y), then

e

is

G.

A b a s i c r e s u l t r e l a t i n g t h e s t r u c t u r e of a t r i p l e system t o t h a t of i t s coiirdinatizing loop is t h e following.

Theorem.

(4)

Let

S = (E,&

e E E,

3-group)

i f and only i f t h e 2-closed subsets of

AG(n,3)

3.

H a l l Matroids

3.

is a group ( i n p a r t i c u l a r , an elementary a b e l i a n

For any

an

I E ~>

be a 3 t e i n e r t r i p l e system,

Ge(S)

f o r some

n

2

E

a r e t h e subspaces of

2.

We may c h a r a c t e r i z e a f f i n e t r i p l e systems

(E,X)

by t h e following

incidence axiom. (A)

If

L = {x,y,z)

E

&

and

w 4 L, then

{w

0

x,w

0

y,w

0

z)

E

de

.

H. P. Young

The proof is obtained at once by verifying that the 2-closure of any basis of

(E,&

has exactly nine points. A proof is also given by

M. Hall [3] for the following equivalent criterion.

Theorem.

(5)

for every point w

S is an affine triple system if and only if

(Ball)

there is an involutory automorphism of S fixing

exactly w.

Now let us consider the implications of Axiom (A) for the cosrdinatizing loop. A

loop G is said to be a Moufang loop if for all x,y,z

E

G

(XY) (2x1 = (x(yz))x Then we may prove: (6)

Theorem. (i)

Let S = (E,&)

be a Steiner triple system.

If S is affine, then the loops Ge(S),

e

E

E, are

Moufang, and they are all isomorphic. (ii)

If for some e

E

E, G,(s)

is Moufang, then S is affine.

Since every Moufang loop has the weak inverse property, (3), (4), and (6) establish a one-to-one correspondence between Hall matroids and commutative Moufang loops of exponent 3 (abbreviated hereafter by "c.M. 3-loops").

Thus we have available Bruck's powerful theory on

cormnutative Moufang loops [2] for analyzing the structure of such systems and for constructing new ones. A basic result is Moufang's

(7)

Theorem. (i)

theorem [5].

(Moufang) Let G be a Moufang loop.

G is di-associative, i.e., every two elements generate

H. P. Young

a subgroup.

.

(ii)

If

(ab)c = a(bc)

for some a,b,c c G' then a,b,c

generate a subgroup of G.

Let G be a cnmnutative Moufang loop. The associator of x,y, and z is the unique element (x,y,z)

of G such that

(xY)~= [x(Yz)~(x,Y,z)

(8)

The associator is skew-symmetric in its arguments, (x,y,z)

(9)

(z,x,Y) = (y,x,z)

-1

and satisfies the relations (10)

(x-l,y,z) = (x,y,z)-l

and (11)

(x,xy,z)

=

(x,y,z)

.

We also have the following useful expansion formula

Z

-

The center of G is the subloop {a

E

G: V x,y E G, ax = xa and

H is normal in G x(yH) = (xy)H, and

( H d G)

(Hy)x = H(yx).

(a,x,y) = (x,a,y) = (x,y,a) = el

if for all x,y

E

G, xH = Hx,

If A 4 G then the quotient loop is

well-defined.

(13)

Theorem.

Moufang loop then

([Z], ~ h .vII, Lemma 5.5) (x,y,(x,y,z))

If G is a comutative

= e, and, more generally,

in the center of the subloop generated by x,y, and z.

(x,y,z)

is

H. P. Young

The Hall matroid found by Hall 133 had 81 points and was constructed by an argument using automorphism groups.

Using our point of view we can

prove that Hall's example is the unique one of this order.

(14)

Theorem.

There.is a unique c.M. 3-loop'on three generators

that is not a group, and it has order 81.

Further, this loop and the

elementary abelian 3-group on four generators are the only c.M. 3-loops of order 81. Proof. -

Let G be a c.M. 3-loop, not a group, generated by

Let h = (a,b,c) )( e, and H = the subgroup

with identity e. generated by h.

By (13),

H d G, so G/H is defined and generated by

aH, bH, cH, which satisfy

theorem (7)

a, b, c and

(aH

bH)

. cH

G/H is therefore a group.

3-group on at most three generators.

= aH ' (bH

cH).

By Moufang's

In particular, G/H is an abelian

If G/H has less than three

generators, then without loss of generality we could write c = h(ab). But then (ab)c = (ab)(h(ab)) Hence, again by (7), IG/H~= 27, and so

= h(a

2b2) = acbc), by di-associativity.

G is a group.

IGI

= 81.

This contradiction shows that

Where Z

(H 5 2) , a similar argument shows that

is the entire center of G

I G/ZI = 27, whence

Z = H.

Further, it follows that any three non-associating elements of G generate G, so (x,y,z)

(15)

E

Z

for all x,y,z

in G.

Hence the expansion formula (12) may be written more simply as follows:

H. P. Young

Since we know, by Hall's construction, that there exists such a loop G, we need only show'that its multiplication is uniquely determined.

,

Letting A = {e,a,aml}

,

B = {e,b,b-'1

, the preceding

C = {e,c,c-l)

shows that every element of G is represented by a member of the set H(A(BC))

= K,

any two elements x = h(xl(x2x3)

9

"

1

0

xx' = hh1hlh2 xl[x;((x;x;)

1

1

9

G.

(x2x3)

Continuing in

I

(x2x3) 1

[ (x;x;)

1

h2 = (xl,x2x3,x2x3)

where and

" I

h3 = (xl,xl, (x2x3) (x2x3) (x;x;)

11

is in the center of

= hh1h1h2h3(x1x;)

For

1

hh9hlx1[ (x2x3) (x1(x2x3) where hl = (x~,x~x~,x~(x x )) 23 this manner we obtain

= 81.

of K we have

and x' = h'(x'(x'xl)) 1 2 3

xx' = hh' [xl(x2x3) 1 [x;(x;x;)

But

IKI

and this representation is unique because

(x2x3) = (x;x2)

(X'3 X 3)

, by

-1

.

di-asociativity, hence

xx' = hh1hlh2h3[(xlx;) [ (x2x;) (x3x;)

II

e

K

.

By (16) and permutations thereof, we find that hl, h2, h3 are uniquely determined by

x and x'.

Hence xx'

is a uniquely determined

member of K. Conversely, let G be any c.M. 3-loop of,order81, and let k(G) be the number of its generators. If k(G) < 3, then G is a group by Moufang's theorem. If k(G) > 3, then a subloop generated by any three elements must have order 2 27, so by the first part of the proof it must

H. P. Young

be a group.

Therefore t h e a s s o c i a t i v e law holds i n

G, s o

and i t is t h e unique abelian 3-group on f o u r generators. case,

k(G) = 3, has been t r e a t e d above.

Corollary.

(17)

The only c.M.

i s a'group

G

The remaining

I

3-loop of order 27 i s t h e a b e l i a n

3-group on t h r e e generators.

t h e unique c.M.

W e w i l l denote by "Gal" t h a t i s not a group, and by v'M81" Let loop of

M

M

is t h e matroid A

t h e corresponding H a l l matroid.

be any Hall matroid, and

based a t

e.

3-loop on t h r e e generators

Ge(M) = G

the co6rdinatizing

The submatroid generated by a s u b s e t

M x c12(A).

i n t h e loop

M, w r i t t e n

i f they a r e d i s j o i n t and contained i n a conmron plane.

to

xH, x

of c o s e t s H.

are parallel i n

is a l i n e of

however, t h a t xlH, x2H E

h

AG(3,3).

x H, x2H 1

€

dHw i l l

Conversely, i f

G.

Suppose

Then f o r any d i s t i n c t l i n e s

xlHl lx2H, because

xlHl

1 x2H

i t can be shown t h a t

A subset of l i n e s of a Hall matroid

p a r a l l e l and p a r t i t i o n

G, but i n

generates a subgroup, and t h e corresponding

AG(3,3), whence

dH , then

partitions

not a l l be pairwise coplanar.

is i n t h e c e n t e r of

ZH,{x1,x2,h}

submatroid i s an

M, and t h e family

G, is p r e c i s e l y t h e family of l i n e s p a r a l l e l

E

The matroid s t r u c t u r e i m p i i e s t h a t

general t h e members of

i n an

H = {e,h,h-'1

G,

L2

G

~~1 I L ~ ,

4

and

"generates"

Two l i n e s (2-flats)

E

L1

A

sense.

h

5G

It is r e a d i l y seen t h a t t h e point-set

u {e) generates M i f and only i f

For any

A

II

is transitive

f o r every p a i r of l i n e s h

i s i n t h e c e n t e r of

M = (E,#)

G.

t h a t a r e pairwise

E w i l l be c a l l e d a c e n t r a l c l a s s of

M.

H. P. Young

A central line is a member of a central 'class. It follows from the above that a line L is a central line if and only if L is contained in the center of the coardinati~in~ loop Gx(M)

for every x e L.

In [3], Hall observes that Msl contains an AG(3,3) matroid.

as a sub-

In any Hall matroid, we shall call a 2-closed subset whose

lines and planes form an AG(3,3)

a

box. By (17), thes'e are just the

2-closed subsets having cardinality 27. Using the present point of view we may state more precisely the structure formed by the boxes in MS1. (18)

Theorem. Let

&'

be the unique central class of the Hall

matroid Msl, and call any two distinct points x,y .of M associates if ~l~((x,~))

E

the boxes contained in M81

first

z', and second associates otherwise.

Then

are the blocks of a partially balanced

incomplete block design on the two associate classes of points.

The

number of blocks is 39, the number of blocks on every point is 13, and

X1

h2

= 13,

Proof. -

Let

=

4.

de'

be the ,unique central class of Msl, and let Q'

the family of planes that contain an element of in.@'

contains exactly three members of &I,

d l .

and

be

Then every plane is an

(dl,@')

affine triple system. For any point e, let H be the unique line in containing e.

Then H

is the center of G,(Msl)

straightforward matter to verify that G' = Ge(M81)/H at H of

(2,O1). Since

that the 2-closed subsets of Let M'

denote this AG(3,3),

G'

is a group and

(oe(,@')

and it is a is the loop based

IG' I = 27, it follows

are the subspaces of an AG(3,3).

which is a rank 4 PMD.

H. P. Young

If M81-

If

P'

M',

then

8(Sf) =

i s any plane of

M',

then

8(P')

has order 27, s o it is a box. Now l e t K

K

be any box i n

@=

Let

K

If

MS1.

is a d i s j o i n t union of elements of

contains no member of

&', l e t

containing

H

e.

subgroup of

Then

G (M

e

a contradiction.

@

)

81

contains any member of

H.

@

.

Mgl

.

M'}

d ' , then

If

K

t h e unique c e n t r a l l i n e

H

Ge(M81) Hence

and

K

Ge(M81)

i s an order 27

is a group,

= HK

Since

Msl.

1@1

M',

= 39..

(and a l s o every p a i r of f i r s t a s s o c i a t e s ) Each c e n t r a l l i n e of

and s o is contained i n e x a c t l y 13planes of

(and every p a i r of f i r s t a s s o c i a t e s )

i n exactly 1 3 boxes. Lx

6

is p r e c i s e l y t h e set of boxes i n

i s contained i n e x a c t l y one c e n t r a l l i n e .

every p o i n t

K

correspondence with t h e planes of

Further, every p o i n t of

M',

hence

d l ,

i s t h e c e n t e r of

@

is a plane of

{@(PI): P'

e e K, and

not containing

Hence

i s i n one-to-one

a point of

UL is 2-closed i n US' is 2-closed i n M81 and

is 2-closed i n

S'

Finally, i f

x

and

y

of

Msl

MB1

is

Hence

M'.

i s contained

a r e second a s s o c i a t e s , l e t

and

L b e t h e d i s t i n c t c e n t r a l l i n e s containing x and y, Y respectively. Then every box of Msl containing { x , ~ ) contains Lx u L hence t h e r e a r e four such boxes--the Y' of M' containing a given l i n e of M'.

(19)

Corollary.

same a s t h e number of planes

contains e x a c t l y 1 3 subloops of o r d e r 27

Gsl

(and they a r e a l l groups).

(19) i s a s p e c i a l case of a theorem of Kulakoff ( s e e [2], Ch. V I , Theorem 3.2),

which s t a t e s t h a t :

if

G

i s a di-associative,

non-cyclic,

n c e n t r a l l y n i l p o t e n t loop of odd prime power order p , then f o r t h e number of subloops of

G

0 < m < n

of o r d e r pm is congruent t o l f p modulo p

2

.

H. P. Young

(Every finitely generated commutative Moufang loop is centrally nilpotentsee below.) Given the existence of Gal, we may immediately construct a c.M. 3-loop of order 3" that is not a group, for any n

4. Namely, we

take the direct product of Gal with n-4 copies of the cyclic group of order 3. Nor are these the only possible constructions. Bruck ([2], Ch. VIII) constructs an infinite c.M. 3-loop containing subloops on k

generators, for each k L 4, that are not factorable as Gal times a group. Next we show that every c.M. 3-loop has order a power of 3. ;

Where A, B, C are normal subloops of the commutative Moufang

loop G, (A,B,C)

is defined to be the subloop generated by all

associators (a,b,c)., where a

E

A, b r B, and c r C.

Then (A,B,C)

is normal in G. The lower central series of G is defined recursively as follows. Go ' G

G is centrally nilpotent of (finite) integer for which G = {el

.

class

n if n is the least

The following fundamental theorem is

proved in [2].

(20)

Theorem.

(Bruck-Slaby)

If G is a comutative Moufang loop

generated by n elements then G is centrally nilpotent of class at most n-1,

H. P. Young

Now let G be a finitely generated c.M. 3-loop with lower central series (el = Gm=Gm-l 5

... = G o

abelian 3-group, hence has order a power of 3.

Since

is a power of 3 and we have proved the following.

Every ~ a l imatroid has 3" points for some integer n,

Theorem.

(21)

and for every n

4.

.

Gi for 0 5 i 5 m-1 and Gi/Gi+l is;aQ elementary

Then Gi+l*

IGI

= G

L

4 there exists a Hall matroid on 3n points.

Configuration Theorems In this section we shall point out a fundamental configuration

theorem that holds in any Hall matroid.

A set {W,X,~,Z) of four noncoplanar points in a Hall matroid, together with the six lines joining them, will be called a tetrahedron, and denoted by

T(w,x,y,z).

T(w,x,~,z)

{W,X,~,Z) generatesan AG(3,3). c12({w of T. (W ?

(22)

y, x

o

y)

0

z))

, and

c12((w

is said to be singular if

The lines c12((w 0

z, x

0

Y))

(X

0

z),

Theorem.

matroid M,

and M'

and

(w

z)

(x

Let T = T(w,x,y,z)

x, y

2 ) )

,

will be called the braces

The midpoints of the braces are the points 0

0

(w

x)

(y

z),

y).

be a tetrahedron in a Hall

the submatroid generated by

{w,xPy,z}

.

H. P. Young

(1) I f

T is singular, then the midpoints of the braces

are identical.

(ii)

If

T i s nonsingular, the three midpoints are d i s t i n c t

and form a central l i n e of

M'.

H. P. Young Proof. Gw(M)

The midpoints of the given tetrahedron may be represented in by

b = y(xz),

a = x(yz),

c

-

z(xy).

They will be collinear if and

only if a(bc) = 1. Now a(bc) = 1 is equivalent to each of the following: -1 -1 bc = a , (bc)c = a c, and bc" bc-I = [(zx)yl

= a ' c .

[z(xY)I-'

But = (~,X,Y)

and a ' c

[x(~z)I-~[(vozI = (x9ySz) follows from the skew-symmetry of the

Thus the collinearity of a,b,c

-

associator (9) in the loop Gw(M). Now let L = Ia.b,cl

.

the unique line containing w

Then v # L, and cl2({w9a-'cl) and parallel to k.

is in the center of the subloop of Gw(M)

But a ' c

L'

is

= (x,y ,z)

generated by x,y,z.

Hence

L', and consequently L, are central lines of the submatroid generated by

Iw,x,y,z)

.I

5. Conclusion In this paper we have investigated the geometric properties of a class of perfect matroid designs, and shown how they relate to the algebra of commutative Moufang loops. This subject is developed more fully in [91. It would be of great interest to find other algebraic structures that give rise to new classes of perfect matroid designs.

H. P. Young

References

Bruck, R.H.: "What is a Loop?" Studies in Modern Algebra, The Mathematical Association of America, 59-99 (1963). : A Survey of Binary Systems. Berlin-HeidelbergNew York: springer (1966).

Hall, M.: "Automorphisms of Steiner Triple Systems". IBM Jour. Res. & Dev., 460-472 (1960). : "Group Theory and Block Designs". Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ. Canberra, 115-144 (1965).

Moufang, R.: "Ziir Struktur von Alternativkzrpern". Math. Ann. 110, 416-430 (1935). Osborn, J. M.: "Loops with the Inverse Property". Pac. Jour. Math. l0, 295-304 (1960). Young, P. and Edmonds, J. : "Matroid Designs". Jour. Res. Nat. Bur. Standards (to appear). Young, H. P.: "Existence Theorems for Matroid Designs". Trans. AMS (to appear). : "Affine Triple Systems and Matroid Designs". Math. 2. (to .appear).