2-Generator golod p-groups

2.

K. A. Zhevlakov, "Alternative rings," Algebra Logika, No. 3, 11-36 (1966); No. 4, 113117 (1967). 3. I. M. Mikheev, "Locally right-nilpotent radical in the class of right-alternative rings," Algebra Logika, i_~i,No. 2, 174-185 (1972). 4. I. M. Mikheev, "Simple right-alternative rings," Algebra Logika, 16, No. 6, 682-711 (1977). 5. R. E. Roomel'di, "Nilpotency of ideals in (--I, 1)-rings with the minimality condition," Algebra Logika, 12, No. 3, 333-348 (1973). 6. E. I. Zel'manov and V. G. Skosyrskii, "Special Jordan algebras of nil-bounded index," Algebra Logika, 22, No. 6, 626-636 (1983). 7. E. I. Zel'manov, "A characterization of the McCrimmon radical," Sib. Mat. Zh., 25, No. 5, 190-192 (1984). 8. E. I. Zel'manov, "On prime Jordan algebras. II," Sib. Mat. Zh., 24, No. I, 89-105 (1983). 9. V. G. Skosyrskii, "On nilpotency in Jordan and right-alternative algebras," Algebra Logika, 18, No. i, 73-85 (1979). i0. V. G. Skosyrskii, " R ight-alternatlve algebras, " Algebra Logika, 23 No. 2, 185-192 (1984). II. A. Thedy, "Ri ght-alternatlve rings," J. Algebra, 37, 1-43 (1975). 12. A. Thedy, "Right-alternative rings with minimal condition," Math. Z., 155, No. 3, 277286 (1977). 13. A. Thedy, "Radicals of right-alternative and Jordan rings," Commun. Algebra, i_~2, 857887 (1984). •

•

,

2-GENERATOR GOLOD p-GROUPS A. V. Timofeenko

UDC 519.45

For every ~ ~ ~ and every field ~ , Golod [1] came up with a construction of a nonnilpotent (infinite-dimensional) ~-generator algebra V over the field ~ , such that every subalgebra with ~ - i generators is nilpotent. Golod's construction yields V in the form q / X , where Q is the free associative algebra of polynomials without constant term in the noncommuting indeterminates ~ , ' ' ' , ~ A over the field ~, and ~ is a homogeneous ideal (i.e., an ideal generated by homogeneous ~olynomials) in 0 • The indicated construction depends significantly on the following condition, to be called the Golod condition, that guarantees that the algebra V "- ~ I X will be infinitedimensional whenever all the coefficients of the inverse of the series

I- g¢ + are nonnegative, where $ ~ ~ generating set of the ideal~ If the algebra V----- Q / ~ algebra.

Let p----- ~ [ p ~

t

and ~ denotes the number of polynomials of degree ~ in a [2, Lemma 3 and Remark]; see also [3, Theorem 26.2.2]. satisfies the Golod condition, it is usually called a Golod

, ? a prime number and V a Golod nil-algebra.

of the multiplicative p-group It is not hard to verify that ~

4+V

The subgroup

is usually called a Golod group.

is an infinite

~-generator

p -group, and if all subalge-

brag with ( ~ - ~ )-generator subgroups in V are nilpotent, then all ( ~ subgroups of ~ are finite [3, Example 18.3.2].

~)-generator

Translated from Algebra i Logika, Vol. 24, No. 2, pp. 211-225, March-April, 1985. Original article submitted May 12, 1984.

0002-5232/85/2402-0129509.50

© 1986 Plenum Publishing Corporation

129

In the present paper (see the proof of Theorem i and Remark 3), we present a method for finding a finitely generated inflnite-dimensional subalgebra of a Golod algebra, which enables us to do the following. i) We can construct a finitely generated nonnilpotent nil-algebra the Golod condition (the example in Sec. 2).

that fails to satisfy

It is meaningful, in this sense, to talk of

a generalization of Golod's result. 2) In connection with V. P. Shunkov's question 6.58 in Kourovskaya Tetrad'

[4], we can

find (Theorem 4) infinite subgroups inside a 2-generator Golod p -group, each generated by a pair of conjugate elements of order an odd prime. the class of 2-generator Golod p -groups ( ~ > ~

Of course, this does not

that in

mean

there are no conjugate biprimitively finite

groups (for the definition, see Sec. 3). First of all, we will prove Theorem 1 for ~ = [ .

The construction of the algebra A

of Theorem 1 was carried out Jointly with V. P. Shunkov for all

i.

~)

~ •

Known Facts, Definitions, and Auxiliary Propositions Let ~

be the free associative algebra of polynomials without constant term in the in-

the subalgebra of F generated by the monomials % .... , ~ Remark I.

with ~ ) [.

The generators we chose for the algebra ~i

have the following properties

(*) and (**): 0

(*) if ~ = ~(~,~)is a monomlal, ~

~and~-~[~

...., ~

then every word in this alphabet that is distinct from $, i.e., V~

$0[~,...~,

.

is also distinct

is the free associative algebra of polynomials without constant term in the

free variables

~i,.,.,~i;

(**) if ~4' ~i

are monomials, one of which has degree distinct from 0, and ~ , ~ £ E

then for every monomial ~ ~ Vl Remark 2. ~"

word in the alphabet ICb 'f

the polynomial

%ii%E

~i '

Vt.

We will adopt the convention that if some polynomial ~ ~ V~ in the variables

is to be considered as a polynomial in the variables

we see, for example, that degree " ~ _---( ~ + ~

-degree

--

~4,...,~5, , then we mark $ with

~.o

Each paragraph that follows will be numbered, and we will refer to the information contanned in the paragraph as a proposition bearing the same number. I. number ~

The Golod condition will be fulfilled if the following restriction applies to the of monomlals of degree ~ in a generating set for the ideal

~ :

(i) where 6 denotes n o w - and for the rest of the article- some positive number ~ 2], see also [3, Example 26.2.3]. 130

J

[I, Lemma

2.

Let ~ be the homogeneous ideal of Golod in the free associative algebra Q

nomials without constant term, in ~

indetermlnates, over the field

resulting infinlte-dimenslonal algebra, e a c h ~ - 4 ~ potent.

p

of poly-

, and let Q/~ be the

-generator subalgebra of which is nil-

The homogeneous polynomials that generate the ideal X then possess the following

properties: a) their degrees run through all thenumbers

~i, N~, Nz+4~., .,~NE,

N~,...~N~, N~+4,

N~+Z,..,~N~, N~+~,... ; b) the number of polynomials of degrees N~,

N +4,...,~N <~=~,[,...] in GoZod's system

of homogeneous generators for the ideal X does not exceed the number

c) we can t a k e f o r N

o

any number s a t i s f y i n g

=m

t h e f o l l o w i n g two i n e q u a l i t i e s :

(2) ;

d) for every polynomial % ~ Q of degree K , the polynomial ~ N ~

(3)

X;

e) each coefficient of a polynomial from the generating set of the ideal ~ is equal to i [3, Example 26.2.5]. 3.

Let ~ ~ [ .

and instead of i : ( ~ 4 ~ ~,,..]

NK

We take ~---- ~ in the construction of Proposition 2; we then put Q-we take

(d.,+4~NK(K~-~,~,...).

As a result, we obtain in ~ an ideal

generated by homogeneous polynomials ~4~Z, ...

that have the following

properties : a) the degrees of the polynomials ~4~ ~Z .... run through all the numbers [~+~N~,

b) the number of polynomials of degrees

(~+~N~,(¢+WbNm,4 ,~,(d.+N)N~(~-4,1,...) ....

does not exceed the numbers

c) for every K = 4,~,..., we have

(4) d) F / ~

is a nil-algebra;

e) all the coefficients of the polynomials $4' ~L'''" are i. Moreover, we will choose each NK

big enough so that the numbers N 4 , N , .,, satisfy

the following strengthening of the restrictions expressed by the inequalities (2) and (4):

131

(5) (6)

4.

In the

construction giveninProposition 2, we take

and we assume that the numbers

,,~ _

~,...

~/.~lt, ~ . ~

, Q I V d ~, ~ - i ~ l ( ~ ,

satisfy, in addition to inequalities

(5) and (6), the following restriction, which is stronger than (3):

.,NK,,, ~l~t@...~'~

~ . Hence, V ~ l ~

.\~','4~Cc[-~ iS an infinlte-dimenslonal algebra, each t~-~1-generator

subalgebra of which is nilpotent.

2.

The Fundamental Theorem THEOREM i.

For each ~ ) ~ and every field ~ , it is possible to construct a 2-genera-

tor nil-algebra A

over the field P , such that ~ contains a proper ~-generator

dimensional subalgebra ~ , and, in addition, all subalgebras with ~-~

infinite-

generators are nil-

potent. Proof.

Let I be a homogeneous ideal of the algebra

= V~/I ~ ~,

F , ~=G/I,

5=(V~+~I/I

. Since

the generating set E of the ideal I must have the following form for the

algebras A and 5 to possess the properties indicated in the theorem:

where the ideal I n

of the algebra

is generated by

polynomials

these polynomials are homogeneous and satisfy the inequality (i) ( ~ nomials of degree ~ among %2~ %0

... ).

and

is the number of poly-

In order to obtain E , we will go through a pre-

liminary construction of a system of homogeneous polynomials

which will generate an ideal g in ~ , such that F / Y

I

~i' ~ ....

We will then obtain

0 0

where the polynomials

is a nil-algebra.

0

(8)

are taken from Proposition 4.

First, we will construct a system of polynomials that we denote like the elements of the set O •

We will then use Lemmas 1-6 to establish that this system is in fact the system U

which, in accordance with (8) above, yields the generating set of the ideal I.

132

U

So, let us start the construction of the system

of homogeneous polynomials.

We will

obtain U from the system ~, ~%,... (Proposition 3) by the following replacement of each polynomial ~ ,

K----4,~,,..by the polynomials ~_4~.~,$&~_~+Z,...,$&~, ~ , &0-----0. Let

~'-- ~+ ~I+...+ N, where M~, "~= 4,[ ....,S, are monomials with coefficient i (property e) of Proposition 3). We may assume without loss of generality that ~%+4' M~;+Z,...,M I t ~ $') are all monomlals

from the set {~,, M~,..., ~, of monomials ~

},

such tat for every M

and ~ , the monomial % 4 ~ - ~

~%f4' Mr+z,.. .°'M s) and every pair V ~ • Then, by definition,

(9) We consider the polynomial

M~+...+ Me. For every ~ ~ we can recall

~e~,rk . , ~ t

(lo)

,

we can find monomials ~ 7~t and % i and the fact that degree > ~(~+~

such. that " ~ ' Z ~ V~ • Hence, (inequality (5)) to conclude

that I

where ~156 V~ , degree ~ <

(i+~

, degree ~ < ( ~ + ~ l

. L~ and ~L are the end and begin-

ning, respectively of some word among 0JI,CSZ,...~O~ or ~ - = ~ or %----4, b m {,~....,~. The same elements (repetitions) in the sets /, ~---~554,~i.....~%] will be denoted by the same symbols ~4 ' ~ ' ' " ~

and ~ = I ~ [ , . . . , ~ % ~

and %,~,...~ ~ ( [ ~ ,

% ~l

, re-

spectively, i.e.,

%, %,..., %, %,...,

L=

By changing the numbering of the elements of {M"4....,~t~

},

. if necessary, we may write poly-

nomial (i0) in the following form:

J,.

/

,

~/

•

,,J

We note that

(n) 133

We will denote the elements of the set

$~.~.~

4~,, ,,

by the symbols

~-4+4'

gK.4+~ ~''','

~hl, ~o~plete, ~he ~onstru~t±on of t~e system U of homogeneous poly, omla~,.

LEMMA i.

If U

is a generating set for an ideal Y

in the algebra F , then

~/Y

is a

nil-algebra. Since ~ / ~ is a nil-algebra {Proposition 3), it follows that for every polynom-

Proof. ial ~

where

~ of degree ~one can find N~

~, ~.are in general

for the ideal [.

such that ~(~+~N~ ~ ~, i.e.,

polynomials with nonzero constant terms, and ~ a r e

generators

It follows from (ii) that each [~is a sum of polynomials belonging to the

ideal Y . Therefore, ~ + ~ N ~ 6 ~ . This proves the lemma. LEMMA 2.

If ~

is a generating set for the ideal I in the algebra ~ , then ~/[ = A

is a nil-algebra. Proof.

It follows from (8) that I = ~

eemma i, ~ / ?

when ~ - [, and [ ~ Y when ~ ) ~ . But, by

is a nil-algebra, which means that ~/I is likewise a nil-algebra, and this

proves the lemma. LEMMA 3.

The ideal I ~ V ~

of the algebra

Vl

is generated by the polynomials

~$~

0

Proof.

Let ~ & [

where B~ and ~

~

. Then ~

[ and

are monomials in ~ , and ~G.L, i-=~,~,...,B%

and ~&~ , i = ~ + 4 ,

I~-?-,...,

~, are polynomials from a generating set of the ideal ~. All the monomials of the polynomial

(12) lie outside the algebra V~ (or else we would obtain a contradiction with the construction (9) of the polynomials ~ 4

~'~, ... ).

Hence, since ~ + V ~ ,

every monomial of the polynomial

(12) must be reciprocally annihilated by some monomial of the polynomial ~ . Let ~ be one of the numbers

<,~, .... F~ and

~ ' ~ - D¢%~4+ D ~ 0 + ' " (~ where

the D:

.

' + T~C%~'~h,'~

. are polynomials in V~, ~ ~---~,~,.,.,£~.

d

f o r some ~ , then none of th~ m o n o m ~ i s of the p o l y n o m i a l •

~G'~5

~L 5

_

(6"~ _ We will prove that if B ~ : ,

~91~ ~"

0' L ~

~,~ V~ i

b e l o n g to the a l g e b r a W

+~deed, D. m ~ - , but ++D, ~+ m V+ • This means that at most one of the monomials ++ lies outside V+ and is di+ti~ct from 1. Suppose, for instance, that ++ ~ < and or 134

a,

a

•

e V~, ~ " 4,~,..:,k, it ~ V~ for ani.~,~. ~,.. , ~ .

V&.

~ ~

-~-i

Since all

So, we arrive

~,

follows from condition

(**)

of Remark i that

at the situation where, after mutual cancellation of those monomlsls of

the polynomial ~b that do not belong to V~ , the polynomial •

=

takes on the form

~'~-~ ~n ~ ~'

This proves the lemma. LEMMA 4.

If K = ~ .~, ....

then the degrees of the polynomials

'I~'--£~,"" will exhaust

all the numbers

(13) Proof.

In terms of the notation used in the construction of the set U , we deduce from

(ii) that 0 ~

degree ~ -

degree ~ L ~ ( ~ + ~ )

that the degrees of the polynomials

$4, ~,...

for ~ _-=~_4~,~.4+~,...~.Hence, are multiples of ~ + ~ ,

by recalling

and applying property

A) of the system ~4,'~~ ... (Proposition 3) we obtain the conclusion of the lemma L E M M 5.

~4~...

does not exceed

Proof. degree

For every K ~ 4,[,... the number of polynomials of degree (13) in the system

~

the set ~ .

As we observed in the proof of the preceding lemma, we have 0 ~ ~(~#~

whenever $~ is obtained from

~

degree ~ L -

in the course of the construction of

This means that in constructing U , the polynomials indicated in Lemma 5 can

only arise from polynomials of degrees

(i+~)N~+ ~.... ,~(~+~)N~,

~ = 4,~,..,

K(~+~)N~+L...

It follows from Proposition 3 (property a), inequality (5)

same number of polynomials of degree (14) in the system

(14)

that there is the

$4' ~Z'"" as there are polynomials

of degrees

in the same system. system

Property b) of Proposition 3 and the inequality (6) assert that the

'4~"-f~'"' does not contain more polynomials of degree (15) than

135

It is obvious that in the course of the indicated construction of the system ~ I' ~'"" the replacement of SK for K "~,~,..,, cannot result in the appearance in this system of more than C~ polynomials. totality.

We choose N

It is not hard to see that the numbers ~

such that N ~ 0~, Km{,~,,.. (for the monomlals

we may take

are bounded in their ~I .... , %

of Sac. 1

This completes the proof of the l e m m a

LEMMA 6. Proof.

The algebra V ~ / [ ~ V~

is infinite-dimensional.

It follows from Lemmas 4 and 3, and the construction in Proposition 4 of the

sequence of polynomials

~

,.., that there are no polynomials in the system

4' Z''"

of degree distinct from

4, N K, N+4, where K = ~,[,°..

where ~K

(16)

From Lemma 5 and inequality (7) it follows that

is the number of polynomials of degree (16) among

0

0

~ , ~ ..... Thus, if

~.

is the

0

number of polynomials of degree ~ among ~4'~Z ..... then

~K=%Nc~+ ~N~"~"%N~t~ +'''+%~N~' Hence, the algebra V ~ / ~ ~ V~

satisfies the Golod condition (Proposition i), and the lemma

is proved. This completes the proof of the theorem. Remark 3. algebra Vi

The subalgebra ~ of Theorem i can be taken to be (Vl +

I /I

, where the

is generated, not by concrete monomials (as in the proof of Theorem I), but by

arbitrary monomials % , .... ~

of degree >, 2 , satisfying the conditions (*) and (**). In-

deed, it is enough to change the proof given in Sacs. i and 2 of Theorem i so as to take account of the change of length of the words ~4''''' 0 ~ over, all the monomials 0J1,...,~

in the alphabet{~,~]

are of the same degree ~ ,

.

If, more-

and are pairwise distinct and

satisfy (**), then it will be enough to replace ~ ar ~ by ~ , and ~ - ~

by ~-{ , everywhere

in Sacs. 1 and 2. Remark 4.

The algebra A will remain a nil-algebra, and its subalgebra ~ will retain

all its properties if, instead of the generating set E of the ideal [ in ~ , we take the set of homogeneous polynomials ~--" £ U ~I' where, for every polynomial ~ ~ ~4 and arbitrary monomials ~I and ~

in ~ , all the monomials of the polynomial %4 ~ ~L lie outside the alge-

bra V~. Example.

Let E ~ = ~

{~),~ ~ 8

~%1

bra ~ . According to the previous remark, algebra.

136

be a generating set for the ideal [ in the algeA " ~/T will then be an infinite-dimenslonal

We will prove that this algebra does not satisfy the Golod condition.

Indeed,

t., r,,

(17)

~=

(18)

Sl

(19) Since, %~=0,

~----~, % ~ = ~ ,

"I,~.=O , it follows that

(20)

$z>~ 0.

(21)

The Golod condition consists in ~ ) O .

~-----0~4,£.....

It therefore follows from (17) and

(21) that (22) (23) from (19), (20) we get that even for $@---[, ~ =

and, since 8 3 ~

, we have S~>~ ~t{%-- S ~ •

0 ,

From this, together with (23), we obtain (24)

6; while it follows from (18) that, even for $~--Z, S = ~ ,

(25) From (22), (24), and (25), it follows that there are no %

for which the coefficients ~,, ~H, ~

are nonnegative. 3.

Finitely Generated Inflnite ~ -Groups We recall that if p - - ~ p ~

, in the construction of the previous section, then I e A

is a multiplicative p -group (p being any prime number).

137

~n thls section, we will always assume thst &--~ + ~ + l - - ~ Z

%--~+?~I=~

+~,

C- - - g r i ~ , ~ .

T ~ 0 R ~ ~. The subgroup ~ --~r(~%~,..., p g ~

of the p -group g - g r ( ~ }

is

infinite. Proof. of ~

Suppose, on the contrary, that ~

over the field ~ ( p ~

is finite.

is a finite group.

Then the group algebra

We will show that the algebra ~

is infinlte-dimen-

signal, and the resulting contradiction will prove the theorem. Indeed, since ~'~-~+[~,~! ~-i z ~-~'; " 0~1~ + ~ ~ (~----4,~,.... ~ , it follows that the monomials -~,----~ ~ ~ -U/~, ~. _ ~ ~ - 4 - ~;...

--

-.J"

=

~.

I

~-~

d

d

nZ~i~0

O

are those homogeneous components of the respective polynomials ~ ~b ~0~,

~'~.....6%~ ~ for which deg ~ - - deg ~ ~ ' " ÷ ' ~ , ~ ' ~ , ~ , . . . a . ,ence, we can construct f o r each polynomial 7~ of the i n f i n i t e - d i m e n s i o n a l a l g e b r a ~ (Theorem 1) a polynomial ~)~ ~

such t h a t % i s the homogeneous component of the polynomial ~ and degree ~ ---- degree

~), i.e., the algebra ~ Remark 5.

is infinite-dimensional.

This proves the theorem.

Theorem i can easily be extended by taking for ~

algebra, where ~ ~ ~

is independent of ~ .

an K -generator nil-

The ~ -generator @ -group obtained through this

construction contains an infinite ~-generator subgroup. In all the constructions below, we will assume that ~ = ~ . By virtue of Remark 4, we can take as a generating set for the ideal I the set ~

=,(P ~, p#~,

p ~ , p-Z, and the analog of Theorem 2 is the following: THEOREM

3.

The subgroup H = g r ( ~ - ~ ,

~ '~g~) of the p -group ~ = gr ~ , ~

is

infinite. COROLLARY i.

The subgroup

~ -----gr (0j,~-~O~ of the

COROLLARY 2.

The ~ -group ~

p -group ~ , L gr ( ~

is infinite.

does not belong to the class of conjugate biprimitively

finite groups for any odd prime ~ . We recall that a group T is conjugate biprimitively finite if for each of its finite subgroups K in N T ( ~ / ~

, every pair of conjugate elements of prime order generates a finite

subgroup. We will construct a group ~

such that it is a Golod group and Theorem 3 remains valid.

To this end, and bearing in mind Remark 3, We take

geneouspolynomials E~=~ U E of ~ r k 4

satisfy inequality (I) for 6 = ~

The set of homo-

4 will be constructed ~ such a way that it will

, i.e.,

%

138

%_~j~,%,=~Z.

~~ ~

(26)

where ~,

is the number of polynomials of degree ~ in ~

i and the convention established in the introduction that

I t follows from the construction of the set E'~ t~t

. We then see from Proposition ~ is a Golod group.

4,~""}~

~!~E,,,.

and Lemm. 5

the number of polynomals of degree E in the set redoes not exceed ~ - - g C ] ~ ' 3 e ~ , ..,

E

~N~4"~, ~N~+I,..o,~4~NK,

-~ ~- ~,Z,..

. From (5) it follows that {~[NI-4) ~ ( 3 - ~ - ~ . ~ ,

i[_~~-~ We will construct the set ~ does not exceed ~ % ~ )

Hence,

z~ ~-~-~ il

in such a way that the number of polynomials of degree I in E

, and the degrees of the polynomials in ~4 are less than the degree

E r ~ ~0" We can take ----(~,~ t" This completes the construction of the se~ t ~ - - E UE~ , and the i . e q u a l i t y (~6) is s a t i s f i e d , i . e . , ~ - - - - ~ (~,~) is a ~ l o d group,

of every polynomial in E • and the following holds. LEMMA 7.

Since ~ - 4 - - ~

l=4~i~b-

~g

, Theorem 3 is valid for a Golod group, and we deduce

from Corollary i that the following holds. LEMMA 8.

The Golod p -group ~ = gr ~ , ~ I

contains an infinite subgroup ~ = gr (~,~-I~I.

It follows from the two preceding lemmas that for all primes ~ > ~ , the 2-generator Golod p-group ~ fails to be conjugate biprimitively finite. In order to obtain the same result for p = 3,~0~, we have to take ~ 1 = ~ g ~ ° ~ ~, 0J~"m ~P~? ~

, and, for the case p - ~ , ~

E~ , by taking ~ = ~ + I THEOREM 4,

I.

we have to modify the construction of the set ~£---So, we have the following:

For every prime ~ # ~

there exists a 2-generator Golod p-group outside

the class of conjugate biprimitively finite groups. The author is deeply grateful to Professor V. P. Shunkov for having posed the problem and for his constant interest in the development of this article.

LITERATURE CITED i. 2. 3. 4.

E.S.

Golod, "On nil-algebras and residually finite p -groups," Izv. Akad. Nauk SSSR,

S~r. Mat., 28, No. 2, 273-276 (1964). E.S. Golod and I. R. Shafarevich, "On the class field tower," Izv. Akad. Nauk SSSR, Ser. Mat., 28, No. 2, 261-272 (1964). M. I, Kargapolov and Yu. I. Merzlyakov, The Foundations of Group Theory [in Russian], 3rd ed., Nauka, Moscow (1982). Kourovskaya Tetrad' [in Russian], 8th ed., Novosibirsk (1982).

139