4-Dimensional compact projective planes with small nilradical

Geometriae Dedicata 58: 53-62, 1995. © 1995 KluwerAcademic Publishers. Printed in the Netherlands.

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4-Dimensional Compact Projective Planes with Small Nilradical Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday H A U K E KLEIN Mathematisches Seminar, Universitiit Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany (Received: 7 October 1994) Abstract. We consider 4-dimensional compact projective planes with a solvable 6-dimensional

collineation group E and with orbit type _> (2, 1), i.e. E fixes a flag v E W, acts transitively on /~v\ {W} and fixesno point in the set W\ {v }. We prove a series of lemmas concerning the action of invariant subgroups of E. These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of E has dimension at least 4. Mathematics Subject Classifications (1991): 51H 10, 51H20.

1. Introduction In the fundamental papers ([16], [17]), Salzmann studies 4-dimensional compact projective planes, i.e. topologicalprojective planes with point space homeomorphic to the classical plane over the complex numbers, We denote by E the connected component of the group of all continuous collineations of a 4-dimensional compact projective plane 7r. The plane is called flexible if E has an open orbit in the space of flags. Since this space has dimension 6 we have dim E _> 6 for such a plane. The group E is known to be a Lie group with dim E < 16. All planes with dim ~ _> 7 are completely classified ([4], [13]). All flexible translation planes (and hence also the dual translation planes) are classified ([2]) and also all flexible shift planes ([1], [11], [12], [3]). By [11] the plane 7r is a shift plane, a translation plane or a dual translation plane if and only if the group E contains a subgroup isomorphic to the vector group R 4. As a general reference for 4-dimensional compact projective planes see Chapter 7 of [18]. In this paper we will consider the remaining case that there is no 4-dimensional abelian subalgebra in ~(E) and dim ~ = 6. If Z is not solvable then ~ --- G L + R ~
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rIhUrd~KLEIN

has an orbit type > (2, 1) in the sense of [5], i.e. ~ does not fix a line Y E L v \ { W } and dually ~ fixes no point u E W \ { v } and acts transitively on Z~v\{W} or on W \ { v } . Up to duality we may assume that Z acts transitively on £ v \ { W } , Next we denote by N the nilradical of E, i.e. the maximal connected nilpotent invafiant subgroup of E. By [14] we know dim N >_ 3. All planes with dim N >_ 5 are known ([6], [7]). By [14], a 6-dimensional solvable Lie algebra with a 3-dimensional nilradical is isomorphic to either 123or to Ic × 12, where 12 denotes the 2-dimensional, nonabelian Lie algebra and Ic is the analogous complex Lie algebra considered as a 4-dimensional real algebra. In order to exclude the possibility g(E) = 13 we consider the base ideal of £(E), i.e. the ideal generated by all 1-dimensional ideals of £(E) ([10, II.5]). We denote the corresponding invariant subgroup of E by B ( ~ ) . The Lie algebra 13 has a big base ideal and in Section 2 we will see that this leads to a contradiction. The most frequently used lemma on the structure of the set of fixed points of collineations q~ E ~ is the lemma on quadrangles: lfq~ E ~fixes a quadrangle, i.e. four points, no three of which are collinear, then q5 = 1. The organization of this paper is as follows. In Section 2 we prove a series of lemmas under the assumption of orbit type >_ (2, 1). These results hold independently of the structure of the nilradical N and they seem to be useful for the general classification in orbit type >_ (2, 1). In Section 3 we describe the subalgebra structure of the Lie algebra Ic × 12. This algebra occurs in Section 4 as the Lie algebra of ~ in the hypothetic case dim N = 3. In Section 4 we will exclude the group of type l c × 12, and thus prove that the nilradical N of ~ has dimension at least 4.

2. General Facts in Orbit Type _> (2, 1) Let 7r = (7~, £ ) be a 4-dimensional compact projective plane and denote by ~ the connected component of the group of all continuous collineations of 7r. We assume the following conditions: • ~r is neither a translation plane nor a dual translation plane, nor a shift plane, i.e. ~ contains no subgroup isomorphic to the vector group R 4. • The group ~ is 6-dimensional and solvable. By [5], ~ fixes a flag v E W. • ~ acts transitively on £ v \ { W ) and fixes no point of the set W \ { v } . In particular, ~, fixes no point in 7:'\{v) and no line in £ \ { W ) . We prove a series of lemmas in this situation. LEMMA 1. Let 1 # N <1 ~. Then N fixes no point in 7~\W andno line in £.\£.v. Ira line l E /2\{W} contains an orbit of N on T ' \ W then v E l, and dually if a pencil £.~(u E P \ {v} ) contains an orbit of U on £.\£.~ then u E W.

PLANES WITH SMALL NILRADICAL

55

Proof Assume N fixes an affine point o E P \ W . Since the group Y]. does not fix the line o V v, there exists another line in £,, containing a fixed point of N, i.e. there is a point o ~ C P \ W which is fixed under N and which is not contained in o V v. Since N fixes the point (o V o') A W E W \ { v } there exists a further fixed point u E W \ { v } . Hence N fixes a quadrangle, so N = 1. The statement about line orbits is proved dually. Assume the line l E £ \ { W } contains an orbit/3 of N in P \ W . Since 1/31 _ 2, the group N necessarily fixes l, and we get v E I. The last statement is proved dually. LEMMA 2. E acts effectively on every orbit on P \ W and on £ \ £ v . Proof Let B C_ £ \ £ ~ he an orbit of E. Denote by N the kernel of the action of E on/3 and apply Lemma 1. L E M M A 3. Let N <1 E be a connected invariant subgroup of E with dim N _< 2. Assume N C_ E[v,~]. Then N C E[,,,w] and if d i m N = 2 then N = E[v,w ]. Proof In the case dim N = 1 the elements of N have a common axis and a common center and these are fixed under E, hence they are equal to W and v respectively. Assume dim N = 2. The set .,4 := { Y E £.~ • N[y,v] ¢ 1} is invariant under the action of E. We have to show that .A = {W}. Assume the contrary, i.e..,4 71 ( £ v \ { W } ) -¢ 0. Since E acts transitively on £~,\{W} this implies £~ \ {W} C_ A. The space P of 1-parameter subgroups of N is a circle and the mapping ~bwhich assigns to a 1-parameter s u b g r o u p / / E P the common axis of its elements is continuous. Since N is the union of its 1-parameter subgroups we have ..4 = ~b(P). Since A contains a subspace homeomorphic to the plane R 2, the mapping ~bcannot be one-to-one, i.e. there are//1, Hz E P with t t l ¢ H2 and q~(H1) = ¢(H2). Hence the group N = (//2, H2) has a common axis. But this contradicts £ v \ { W } C_ ¢4. The last assertion is clear. L E M M A 4. Let N ~ E be an invariant subgroup with non-trivial centralizer C~ N ~ 1. Suppose that N consists of perspectivities.

1. Let z and A be the center and the axis of an element a E N \ { 1 } respectively. Then v E A and z E W. 2. N C E[w,w] or C~.N C_ E[v,,,]. Proof (1) Apply L e m m a 1 to the invariant subgroup C~N. (2) Assume N ~ 1. Let .A be the set of all axes of elements ¢ 1 of N. By (1), .,4 C_ £~, and .A is invariant under the action of E, hence ,4 = {W} or £ v \ { W } C_ ~4. In the first case we have N C E[w] and by an application of (1), N C_ E[w, wI. Now assume £ v \ { W } _c ,4. The centralizer C ~ N fixes the elements of the set JI, i.e. C ~ N C_ E[v]. A final application of part (1) to Cr.N yields C ~ N C_ E[v,v]. LEMMA 5. Let N <1 E be an invariant subgroup and assume C~.oN ~ 1for some affine point o E P \ W. Then N C_ E[v,~].

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Proof. Assume N ~ 1. The centralizer Cr.o N fixes each point of the orbit ON. Assume oN is not contained in a line. Then there exists a point oI E oN\(o V v). S e t b : = ( o V o ' ) A W E W \ { v } , X := o V o I = o V b a n d Y := o V v . Since oN ~ X there exists a point 0I' E o N \ x . Since Cr.oN fixes the points o, v, b, 0'1 the lemma on quadrangles implies o" E Y. But then Cr, oN fixes the quadrangle o, v, b, (o"Y b) A (01Y v); a contradiction. This shows that oN is contained in a line and Lemma 1 yields oN C_ o V v =: Y and y N = y . Since E acts transitively on £ ~ \ { W } this implies N C_ Ely]. By Lemma 4, N C_ E[v,v]. LEMMA 6. The center of E consists entirely of perspectivities with center v and axis W, i.e. Z ( E ) C_ E[,~,w]. Proof. Since Eo ~ 1 for each point o E P \ W , Lemma 5 implies Z(E) _C E[,,~,]. Since the axis of an arbitrary element ff E Z ( E ) \ { 1 } is fixed under E, the axis of such an ff is W, i.e. Z ( E ) E E[~,w]. LEMMA 7. Let N ~_ E be a 1-dimensional connected invariant subgroup. Then N C__Ely,v]. Proof. Choose an affine point o E P \ W . Since dim Eo > 2 and dim C ~ N >_ 5, we have Eo N C ~ N ~ 1. Lemma 5 implies N C_ E[~,~]. LEMMA 8. We have B ( E ) C__E[v,w] and, inparticular, d i m B ( E ) _< 2. Proof. By Lemma 7 each connected 1-dimensional invariant subgroup is contained in E[v,v] and Lemma 3 implies that these groups are even contained in E[v,w].

3. The Lie Algebra 12c × lz

We consider the 6-dimensional solvable real Lie algebra G = IC2 × 12. The righthand factor 12 is given as 12 = (e, f) with [e, f] = e. The left-hand factor 1c may be generated by the matrices

These matrices satisfy [R,S]=[B,C]=O,[B,R]=B,[B,S]=C,[C,R]=C

and [ C , S ] = - B .

The Lie algebra ~ has exactly one 1-dimensional ideal, namely (e) and exactly two 2-dimensional ideals, namely (e, f) and (B, C). In particular, the full automorphism group r of G must fix the real 12, and consequently it must fix the centralizer of this subalgebratoo, i.e. 12 c is invariant under F. This implies F = Ant(/c) × Ant(/z).

PLANES WITH SMALLNILRADICAL

57

The automorphisms of 12 are e ~ ae, f ~ / S e + f with a , 8 E R, a ~ 0, and such an automorphism is inner if and only if a > 0. We describe the automorphisms of the (real) Lie algebra 12c. Write (z, w) for the element a R + bS + cB + dC of 12c with z = a + ib and w = c + id. The inner automorphisms are the mappings (z, w) ~ (z, za + wb) with a, b E C, b ~ 0 and the outer automorphisms are (z, w) ~ (2, 2a + ~b) with a, b E C, b ~ 0. By a routine calculation it is possible to determine all subalgebras of G up to the action of F. The resulting list is given in Table I. The ideals of G not containing G' are exactly: (e) = l~, 12, (B, C) = (/2)c,,l 2c and those subalgebras of 1c which contain (lC) '. The simply connected Lie group with Lie algebra l c is the group L c := {(z,b)- z,b ~ c}, with multiplication (Zl, bl)" (z2, b2) = (Zl + z2, eZZbl + b2). The 1-parameter subgroups of this group corresponding to R, S, B, C are given by: R(t) = ( t , O ) , S ( t ) = (it, O ) , B ( t ) = (O,t)

and

C(t) = (O, it),

respectively. The center is Z = {(2~in, 0): n e Z}, and consequently the Lie groups with Lie algebra l c are enumerated by a natural number n E N U { ~ } . We write L 2c , ~ = L c and for each n C N: L 2,~ c = {(z,~). z e c \ { 0 } , b e c } with multiplication

(zl, bl). (z2, b2) = (z1~2,461 + b2). In particular L 2,1 c is the ordinary complex L c. 4. Excluding a 3-Dimensional Nilradieal

In this section we will prove the following theorem: THEOREM. Let 7r be a 4-dimensional compact projective plane and denote by the connected component o f the group of continuous collineations of Tr. Assume

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HAUKEKLEIN

TABLE I.

2-dimensional subalgebras abelian (aR + bS, f) ((a,b) E R2\{0}) (B, f) (~n + bs, ~) ((a, ~) e S:\{0}) ( b S + f , R + b ' S ) (b,b' eR, b¢O) (~n + bs, R + ~) ((~, b) e S:\{0}) (S, R + f) (B, ~)

non-abelian (e, aR+ bZ+ f) (a,b ~ R) (e, B + f) (B, R + yf) (y E R) (B,R+~) ( B + e , n + f)

(R,S) (B, C + ~) (B+yf, C+flf) (B+e, cB-I-aC)

(y, f l e R) ((c,d) eR2\{0})

3-dimensional subalgebras (R, S, F) (B,R,F) (aR + bS, e, f) ((a,b) e R2\{0}) (R+flf, S-Ffl'f,e) (fl, fl' e R) (R+flf, B,e) (tiER) (B, ~, f) (B,n-t-ae--t-f,C-t-e) (c~ e R) (aR -F bS -F e, B, C) ((a, b) • R2\{0}) ( a R + b S + f , B , C ) (a,b E R) (B + ~, ~B + de, R + ~ + f) (~ ~ R, if, d) • a~\{0}) ( B , R + f l f , e) (fl eR, fl # 1) (B, dC + flf, e) ((d, fl) e R2\{0}) (B,C, an + bS) ((a,b) e R2\{0}) 4-dimensional subalgebras not containing ~' (R, s, ~, f)

(R,B,e,y) (R+o~e,S+oz'e,B,C) (c~,(~' C R) (aR -1-bS, B, C, f) ((a, b) e R2\{0}) (R+flI, S + f l ' f , B , C ) (fl, fl' e R) (B, C, aR + bS, e) ((a, b) e R2\{0}) 5-dimensional subalgebras Up to the action of I', there is only one 5-dimensional subalgebra not containing ~', namely (R,S,B,C,f).

PLANES WITH SMALL NILRADICAL

59

that 7r satisfies the assumptions of Section 2 and denote by N the Nilradical of ~. Then dim N _> 4. Assume that we are in the situation of the theorem but dim N < 4. We will show that these assumptions lead to a contradiction. LEMMA 1. We have g(S) TM 1c2 × 12. Proof Since d i m U _< 3 we have g(~) ~- 12c × 12 or g(~) ~ l 3. The latter possibility is excluded by Lemma 8 of Section 2. Lemma I implies that S = L 2,~ c × L2 for some n E N tO {c~}. LEMMA 2. E acts transitively on P \ W, and L~2 C_ S[~,w]. The invariant subgroup C i (L2,n) has a trivial centralizer in So for each affine point o E P \ W. Proof By Lemma 7 of Section 2 we know that L~ C S[~,w]. Let o E 79\W be an affine point. If C~o(L~n )' ¢ 1, then (L2,n) c , C ~[v,v] by Lemma 5 of Section 2. But then Lemma 3 of Section 2 implies that (Lz,n) c t = E[v,w], contradicting the fact that U2 C_ Sly,W]. If S is not transitive on P \ W , then there exists an affine point o E 79\W with dim So _> 3. By an inspection of the list of subalgebras of 12 c × 12 given in Section 3, we see that (Lz,n) c I = exp((B, C)) has a non-trivial centralizer in So; a contradiction.

LEMMA 3. Sly,W] = L2 and n = 1, i.e. S = Le2 × L2. Proof Choose an affine point o E P \ W . By Lemma 2 the stabilizer Eo is a 2-dimensional connected subgroup of E which contains no invariant subgroup of E, and the centralizer of (Lz,,~) c in Eo is trivial. By the list of 2-dimensional subalgebras of 12 c × 12 of Section 3 there are four possible cases for the subalgebra

g(Zo) (up to an automorphism of 1c × 12). These cases are:

1. (bs + f , R + b'S)(b, b' e R), 2. ( a R + b S , R + e ) ( ( a , b ) E R2\{0}), 3. ( S , R + f),

4 (a, s>. In any case, CEoL 2 • 1 and Lemma 5 and Lemma 3 of Section 2 imply L2 = ~[~,w]. By L e m m a 6 of Section 2, we know that Z(LC2,,~) C_ Z ( S ) C_ E[v,w] = L2, hence Z ( L ~ ) = 1, i.e. n = 1. Choose o E 79\W and let Y := o V v E £ ~ \ { W } . We choose the coordinates in g(S) in such a way that e(So) is one of the four subalgebras in the proof of Lemma 3. If we let C := C~oL2, then we have seen that d i m C > 1, and obviously C C_ L2c f3 S[r-]. The group M := exp(R, S) is 2-dimensional abelian with C M M # 1, hence M C S y . This implies S y = M × L2 i.e. g ( S v ) = (R, S, e, f ) . We describe L2c in the coordinates of Section 3, i.e. L c = { (z, b) : z E C\{0}, b E C}. In these coordinates we have M = {(z, 0)" z E C\{0}}. The

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HAUI~ KLEIN

group L2 is described similarly as L2 = {(s, t) : s, t E R} (with multiplication

= (,1 + , 2 , e %

+ t2)).

L E M M A 4. E acts transitively on Wk {v} and £\£v. Proof Assume to the contrary that there exists a 1-dimensional orbit B of E on W \ { v ) . Since C C E[y], there exists a b E B with C ~ Eb, hence L c g Eb. By the list of 5-dimensional subalgebras, a 5-dimerlsional subalgebra of 1c × 12 which does not contain the subalgebra Ic contains the commutator of 1c x 12. This implies Eb = E[B] __DE'. Since L2 = E[v,w] is a regular normal subgroup of the action of Eb on L b \ { W ) , we get E[b] # 1. Since each element of Eb fixes B pointwise, we know E[b] = E[b,W] C_ E[w,w]. Therefore the group E[w,w] must be abelian ([15, 8.1 ]), but it contains L2; a contradiction. A dual application of Lemma 2 yields the last assertion. Let K := (LC)' = {(1,b): b E C} and A := K x L2 _<3E. The above calculation of E y implies that the group K acts regulary on £ . \ { W ) . Hence A acts regularly on P \ W and by duality A acts regularly on / : \ ~ . , too. Again by duality the stabilizers EF and Eb for an arbitrary point b C W k { v } are conjugate in E, hence E y fixes a point b C W \ {v}. Let X := o V b E £\£~. L E M M A 5. For each line A q / : . \ { W } there exists exactly one point ¢(A) C

W\{v} with EA = E¢(A) and the mapping (9 : £ v \ { W } --* W \ { v ) is a bijeclion. Proof We have proved E y = Eb for some point b E Wk{v}. Assume that there is another point b~ C W \ { v , b} with E y = Eb,. Since Eb = E y = M x L2, we have E[w] = L2. The effective factor group E/E[w] --- L c has the subgroup M as its stabilizer at the point b, i.e. the action of E / E [ w ] on W \ { v } is equivalent to the natural action of L2c on C. Hence E y = Eb,b, = E[W] = L2; a contradiction. Since E is transitive on £.~\{W}, this proves the first statement. By duality we get a map ~b : W \ { v } --+ f_.~\{W} inverse to ¢. Coordinatize P \ W by A. Then £ ~ \ { W } = {{p} × L2 : p e K}. L E M M A 6. Eo ~ R x SO(2) acts regularly on £o\{Y, o V ¢(Y)} and on

&\{Lw}. Proof Let Y' e £v \ { Y, W ). The group Eo,y, fixes the quadrangle v, o, ¢(Y), Y' A (o V ¢(Y')), hence Eo,y, = 1. Consequently Eo acts regularly on the cylinder E~k{Y, W ) and in particular Eo - R × SO(2). The lemma on quadrangles and L e m m a 5 imply the regularity of Eo on £o\{Y, o V ¢(Y)}. By our choice of the coordinate system in l~ × 12 there are the following two possibilities for the subalgebra ~( E o): • Case A: g(Eo) = (R + bf, S + df) with b, d e R. • Case B: e(Eo) = (R q- ae, S + ce) with a, c e R.

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CASE A: We get

So

d

=

z,0) : z E C)

where 9~z and 3z denote the real and imaginary part o f z E C respectively. Since So is isomorphic to R × SO(2) we get d = 0. Since So acts regularly on £ v \ { Y , W } the group So must act regularly on X\{o, ~(Y)), too. The orbits of Eo on A which are not contained in L2 are given by

13s,t = {(1,e~,s, eb-~zt): z E C )

with

s,t E R.

Choose ~, t E R with X \ { o, q~(Y) ) =/3,,t. Since o is in the closure of X \ { o, ~b(Y) } we get s = 0 and t = 0 or b > 0. I f t = 0 then X\{~b(Y)) = K × {1}. But the transitivity of A on £ \ £ v implies/2\£,, = { K × {q) : q E L2}; a contradiction. Hence we get b > 0 ~ t. In this case: X\{~b(Y)) = {(1,eZ, O, teb-mz): z E C} tO { ( 1 , 0 , 0 , 0 ) } = { ( 1 , e L O , tle~lb) : z E C} U { ( 1 , 0 , 0 , 0 ) }

= {(1,z,O, tlzlb):z

E C}.

For each z E C we get the line

X(z)

{(1,w,O, tlw- zl b- tlzlb).w E C).

: = X (l'~,°'-tH~) =

Since A acts regularly on f-.\£v these lines are pairwise distinct. All lines X ( z ) pass through o and since the map z ~ s ( l l - z lb - Izl b) is not one-to-one we get two complex numbers z, z J E C with z # z' such that the distinct lines X ( z ) , X ( z t) intersect in at least two points; a contradiction. CASE B: In this case we have

Eo = {(eZ,O,O,a~Rz + c3z) : z E C). Since Eo ~ R × SO(2) we get c = 0. The orbits of Eo on A which are not contained in L2 are given by =

+

-

:

E C).

As in Case A we can choose s, t E R with X\{o, ~b(Y)} = Bs,t. The condition o E B~,t implies in turn: s = 0 and t = 0. This shows X\{~b(Y)} = K × {1) and, as in Case A, this immediately implies a contradiction. Now all cases have led to a contradiction and the theorem is proved.

References 1. Betten, D.: Komplexe Schiefparabelebenen,Abh. Math. Sere. Univ. Hamburg48 (1979), 76-88.

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2. Betten, D.: Zur Klassifikation 4-dimensionaler projektiver Ebenen, Arch. Math. 35 (1980), 187-192. 3. Betten, D. and Knarr, N.: Rotationsflachenebenen, Abh. Math. Sere. Univ. Hamburg 52 (1987), 227-234. 4. Betten, D.: 4-Dimensional compact projective planes with a 7-dimensional collineation group, Geom. Dedicata 36 (1990), 151-170. 5. Betten, D.: Orbits in 4-dimensional compact projective planes, J. Geom. 42 (1991), 30--40. 6. Betten, D.: 4-Dimensional compact projective planes with a nilpotent collineation group, Mitt. Math. Ges. Hamburg 12 (1991), 741-747. 7. Betten, D.: 4-Dimensional compact projective planes with a 5-dimensional nilradical (to appear). 8. Betten, D. and Klein, H.: 4-Dimensional compact projective planes with two fixed points (to appear). 9. Betten, D. and Polster, B.: 4-Dimensional compact projective planes of orbit type (1,1) (to appear). 10. Hilgert, J., Hofmann, K. H. and Lawson, J. D.: Lie Groups, Convex Cones and Semigroups, Oxford, 1989. 11. Knarr, N.: Topologische Differenzenfl~ichenebenen, Diplomarbeit, Kiel, 1984. 12. Knarr, N.: TopologischeDifferenzenW, ichenebenenmit nichtkommutativerStandgruppe, Dissertation, Kiel, 1986. 13. L0wen, R.: Four-dimensional compact projective planes with a nonsolvable automorphism group, Geom. Dedicata 36 (1990), 225-234. 14. Mubarakzjanov, G. M.: On solvable Lie-algebras, lzv. Vyssh. Uchebn. Zaved. Math. 1 (1963), 114-123. 15. Pickett, G.: Projektive Ebenen, Springer, Berlin, 1975. 16. Sa•zmann• H. R.: K•uineati•ns gruppen k •mpakter 4-dimensi•na•er Ebenen•Math. Z. • • 7 ( • 97•)• 112-124. 17. Salzmann, H. R.: Kollineationsgruppen kompakter 4-dimensionaler Ebenen II, Math. Z. 121 (1971), 104-110. 18. Salzmann, H., Betten, D., GrundhSfer, T., H~ihl, H., L6wen, R. and Stroppel, M.: Compact Projective Planes, de Gruyter, Berlin, 1995.