Width of verbal subgroups in solvable groups

WIDTH OF VERBAL SUBGROUPS IN SOLVABLE GROUPS V. A. Roman'kov

UDC 519.4

i. Let ~ be an arbitrary group and ~f~ the verbal subgroup of ~f .

We will say that Uf~

has width

product of ~values of the words If such an ~ does not exist,

~I

~ if every element ~ in G ,

~ , defined by a word

~f~ can be represented as a

and ~ is the smallest number with this property.

we will say that the width of the subgroup

~

is infinite.

The question of finiteness of the width of the verbal subgroup comes up in various group theoretic problems.

For instance, the condition that the subgroup

implies that the corresponding verbal subgroup

Uf~

gF~

be of finite width

of the profinite completion ~ of the

group ~ is closed; moreover, a verbal subgroup of finite width is formal, etc.

M. I. Karga-

polovhas posed in [i] the following problem (4.34) for solvable groups: "for which words ~f is the width of the verbal subgroup Uf~ of every finitely generated solvable group ~

finite? Would

not this be the case for the word ~ f = [ ~ ~] ?" The concept of width (in different terms) stems from P. Hall, who considered it in connection with the question: values?"

"will the subgroup

~f~

be finite if ~f has a finite set of

The concept of width appears, clearly spelled out, in a thesis of Stroud (a student

of P. Hall), which he defended in 1965.

Unfortunately, Stroud's early death (he died shortly

after receiving his doctorate) prevented him from publishing the proofs of his thesis, and these results remained practically inaccessible.

the results in

The term "width of a verbal

subgroup," used in this paper, was introduced by Merzlyakov [2], who established that every verbal subgroup of an algebraic matrix group has finite width. We proceed with some basic known results about the width of verbal subgroups. Let

~

be the blass of all Abelian groups, ~

nilpotency degree K; on normal subgroups.

~

the class of all polycyclic groups, ~

We will let ~

i. (Stroud, Unpublished). ~

If

the maximal condition

denote the product of classes of groups; i.e., the

class of extensions of groups from ~ by groups from ~

subgroup

the class of all nilpotent groups of

~£~K

.

is a finitely generated group, then every verbal

is of finite width.

2. (Rhemtulla [3]). Every proper nontrivial verbal subgroup of the free product of groups ~ .

I~I~.

I~I >~2 , has inifite width.

3. (Rhemtullah [4]).

If ~ £ ~

is a finitely generated solvable group, where ~ i s

a class in which every finitely generated group satisfies ~ X ~ with respect to the word

, then ~l has finite width

~f= [~,~]°

By a well-known theorem of P. Hall, finitely generated groups from the product ~ satisfy ~ L ~

hence, the above assertion is valid for finitely generated

a

~-groups.

Translated from Algebra i Logika, Vol. 21, No. i, pp. 60-72, January-February, 1982. Original article submitted August 25, 1980.

0002-5232/82/2101-0041507.50

© i982 Plenum Publishing Corporation

41

4. (Wilson

[5]).

If

~E~

and Uf is an outer-commutator word,

then

~

has finite

width. In the present article we remove the restriction on the word ~f in Wilson's

theorem,

and thus give an answer to his question in [5]. THEOREM i.

If

G E~

and Uf is an arbitrary word,

then the verbal subgroup

~f~

has

finite width. In the course of the proof of the theorem, we will also obtain some results of independent interest.

We will first establish

of a finitely generated nilpotent group

that the width of an arbitrary verbal subgroup N,

is finite.

For this purpose,

~f~V

the concept of a

verbal subgroup, or, more precisely of the word ~ , will be somewhat broadened.

This will

enable us to prove the finiteness of the width of an arbitrary verbal subgroup of an almost nilpotent group, and also of a group that is representable as a product of a finitely generated nilpotent normal subgroup by a finitely generated almost nilpotent subgroup. sequel, we will make use of a theorem of Zaitsev contains a subgroup of finite index

In the

[6], which states that every polycyclic group

of the type just described.

The basic difficulty

consists

in carrying over the statement of the theorem from the subgroup of finite index to the group as a whole.

In order to do this, we will pass to the covering polycyclic group, which is

representable

as a product of the nilpotent radical by an almost nilpotent subgroup.

It will

follow from the results obtained that every verbal subgroup of the covering group has finite width;

and the homomorphism is easily seen to preserve this finiteness.

The remainder of the article is devoted to the construction of a series of examples of verbal subgroups of infinite width.

Hartley showed in [7] that every subgroup of finite index

of an inverse limit of finite solvable groups of bounded Fitting length is closed, proved that the inverse limit is finitely generated in the topological sense.

His proof relies on a

proposition, which he also proved, and which states that the width of the commutator subgroup of a finite solvable group is bounded from above by a function of the Fitting length and the number of generators of the group. the profinite

completion,

analogous statement THEOREM 2.

It follows from this that the commutator subRroup of

of an arbitrary finitely generated solvable ~ronp, is closed.

for arbitrary verbal subgroups is false,

If

~

,~

, and

subgroup

~'~z) is not closed in the inverse limit topology.

2. ~ - V e r b a l

~

is its pro-~-completion

/fIX)

Let

{~,~Z,,.,,~,...}

let ~ )

The

elements

phism

; and

consider

~--~Au~ ~ ~X}~

the

of a

free

~E~]

fixed

be

product

will

map,

be the free group of countable

be

the

the

free

~[~]

called

image

lies in ~ .

called its value in the group

~

group

~=~{X)~[~) ~-words.

of which

can then be uniquely extended

of the subgroup

42

then the second commutator

the generalization of the concept of verbal subgroup plays a

purely technical role,

us

for some /Z

containin~ the

Subgroups

In the present article,

Let

as the following theorem shows.

~is a free group of rank 3 in a variety of groups,

variety

The

will

rank on the set X =

on

the

set ~

and

its

subgroup

Let ~ also

be

be

an

free

bye.

~ ~,H0~. ~

All the values of a qg-word

generators.

~[~]=~{X)~F~

arbitrary

denoted

to a homomorphism

The image of a ~ - w o r d

of

group,

• and

Every homomorThe image under

under such map will be Uf in ~ ,

taken toRether,

generate a ~ vious way.

-verbal subgroup ~f~,

In the case where

Let ~ be a group, ~ subgroup of ~ .

for which the concept of width is defined in the ob-

uf6~{X), we get back the usual definitions.

a set of automorphisms of G

The width of any ~ - v e r b a l

of the subgroup Uf~.

subgroup

, and A 4 ~ uf~/A

a ~ - a d m i s s i b l e normal

cannot then exceed the width

In the future, wewill denote widthby r ]~f and the previous statement

then reads ~Uf~ A]~f ~ E~f~]~, The group @-word

~[~]

g~'6~ [ ~ ]

itself admits the group of automorphisms F ~ ) . , one can define the ~ - v e r b a l

Hence, for every

subgroup ~Y'~-[~], generated by the closure

of ~#" with respect to all those endomorphisms of the group ~ that are the identity on ~ ( ~ ) and arbitrarily map

~(~)

sidered to be finite.

into itself.

~

of ~ff, that

~f is an element of the subgroup ~ [ ~ ] ~ f = , where the set X~-~{~;,~2 ...., ~ }

of the group ~ a ~ = ~ X ~ ) ~ ( ~ }

tains all the variables that enter into the word ators of

~ , and

~r

in this context, ~

may be con-

Moreover, if ~ is finitely generated, we may assume, in the defini-

tion of the ~-verbal subgroup

~-(X~)'E~(Xl;},~t'(~)]

It is clear that, in this context, ~

~f , /~

consists of all the elements o f ~

belongs to some variety of groups ~ ,

con-

is not less than the number of generthat appear in the ~ - w o r d ~ .

If,

~(~] will be considered to be free

in this variety. 3. Verbal Subgroups of Nilpotent Groups The width of an arbitrary subset ~ defined in the natural way. LEMMA 3. i. ~:~ -wo r d ,

If

A~

of a ~

-verbal subgroup

~f~

of some group ~ is

The following lemma then holds. is a ~ - a d m i s s i b l e normal subgroup of a group ~ , and ~

is some

then

E C/A] P r o p o s i t i o n 3.2.

automorphis~ o f N ,

Let

~d

N~ ~K

+

be a f i n i t e l y

ur an a r b ~ t r a ~

n

generated n i l p o t e n t group, ~

~ -word.

~en

the

~-verbal

some s e t o f

subgroup

~/

has

f i n i te w i d t h .

Proof.

Let us consider the free product F~r'~{~)" ~(~U$) , where

nilpotent group of degree of ~

PC of sufficiently large rank, and ~ r

that enter into the expression for ~r.

of the quotient group

The word

lower central series of a gropu H .

is the a free

is the set of elements

~r can be considered as an element

FC~J~-FE~3~//,+,F[~3~ where ,

~{X~)

I~H

denotes the i-th term of the

we have already pointed out that it is sufficient to

establish that the ~-verbal subgroup

~f/E~]~

has finite width.

We represent the word

as a product of extended commutators in the images of the generators of the group ~ ( X ~ ) and of elements of the form contains the elements

~+!

~

.~6

Let us suppose that each of these commutators

at most ~ times.

We will consider the subgroup ~(~) of ,~-[~]~f

generated by all those commutators in the images of the generators of the group ~ ( X ~ ) ~ -~/ and of elements of the form @ ± / , ~ 6 ~ r

that contain the latter at most ~ times.

obviously a finitely generated nilpotent group.

~(~) is

We can run the central series

K(~ = ~'c~)t > ,kE~ ~....

~ Kc~,~ ~- I,

(I)

43

through it, where

K(~)~ is generated by all those commutators that contain at least ~ entries

of images of elements of X~ Since the group Let

Ul /7~ .....U S

K(~) is Noetherian, its subgroup

be generators for it.

K(~)K has finite width.

U/K(~)NK(~#~

is finitely generated.

We may, by an induction argument, assume that ~T~)I

We use here the fact that series (i) admits those endomorphisms

that fix ~ul and are extensions of endomorphisms of the group ~tX~) . we observe that series (i) can be completed to a ~ UF~(~) ~(~)K

-admissible series in F[~Ju/"

is also of finite width.

We will prove that the subgroup

We consider the endomorphism

&:

~[~]~---~]U/,

which is defined to be the identity on ~ and defined on X~ according to the rule ~ --~ n7 ~ ~m&. We get an induced map ~ - - ~ , ~'=/,2,...,$ . By a well-known theorem of Hilbert (the solution to Varing's problem), there exists a natural number Z = ~(K)

such that every

natural number rr& can be written in the form

fTZ=~K~/TZ~+0 K Let ~. , ~ 0.+/TZZo J be the width of the element ~,, ~/,£ ..... $ ; it is then obvious that [ufK(~) n ~({)K] ~ ~ ( ¢ + ~ ~ + ~ ~ It remains to observe that

~{~j

dY-~E~3~/, and, by Lemma 3.1, the width EU/~({)]U/=

is finite.

COROLLARY 3.3. subgroup

~f~

Proof.

Let G be a finitely generated almost-nilpotent group.

Then every verbal

has finite width. The proof is carried out by the method used in [2]. Let ~,,~a ..... ~

be a

system of representatives of the comets of the group C by the a nilpotent normal subgroup of finite index.

We consider a finite collection ~ o f

if O/ is a word in /g variables. by the elements f/'~z'""~ " by the set of words ~/~.

We assume here that the automorphisms in ~ are conjugations

By Proposition 3.2, the ~

V , has finite width.

COROLLARY 3.4. Let mal subgroup of & ,

defined

~f~/~

is finite.

According to

is finite.

~ = ~

and M

-verbal subgroup M ~

The word 6~ is finite-valued in the quotient

By a well-known theorem of P. Hall, the subgroup

Lemma 3.1, the width [ ~ ] a f

V -words of the form

be an almost-polycyclic group, where N is a nilpotent nor-

is an almost-nilpotent subgroup of ~ .

The verbal subgroup

W&

is then of finite width. Proof.

We consider the set V of words of the form

where 9~E~,

and

~

are variables, which we allow to take on values in the group ~ .

~

be the ~

-verbal subgroup defined by these words in ~ .

V~

is finitely generated,

position 3.2, V ~ the word ~gr in ~ has finite length.

Since group ~ is Noetherian,

and we need only consider a finite set of words in V

has finite width.

We now pass to the quotient ~l=~/V/%/

lie in the image of the group ~

Let

By Pro-

The values of

hence, by virtue of Corollary 3.3, ~f~

Lemma 3.1 now yields the desired conclusion,

4. Verbal Subgroups of Polycyclic Groups Let ~ be a polycyclic group and ~ its nilpotent radical.

44

By a theorem of Zaitsev [6],

one can find a nilpotent subgroup ~

of ~- such that the product ~----N~ has finite index in O

G

We may assume the subgroup T to be normal in ~ .

that every verbal subgroup UfT has finite width.

Corollary 3.4 immediately establishes

We observe, however, that the verbal sub-

group of a subgroup of finite index can be quite different from the corresponding verbal subgroup of the group as a whole.

For instance• the commutator subgroup

( N o ' Z ) I of the holo-

morph of the infinite cyclic group Z is isomorphic to Z , while Z t = 0 ,

Hence, before we

pass to the proof of Theorem i, we will alter slightly the above-mentioned decomposition of D. I. Zaitsevo LEMMA 4. i.

Let ~ be a polycyclic group, ~ =

of some normal subgroup of finite index in ~ . homorphism ~=~t

~:~

on ~

such that ~

NM

an arbitrary Zaitsev decomposition

There exists a polycyclic group

~I and a

contains a normal subgroup ~ of finite index, with

its analogous decomposition, such that the following conditions are satisfied:

Proof.

&=T.

We first consider the case

natural homomorphism ~ : Y

* ~.

of all elements of the form

~-I

is a nilpotent normal subgroup in is isomorphic to ~

.

We take the free product ~ - - ~ ' ~

Then ~ [~,~] ~ ~

.

Let

(where ££ ~ , a E ~ ) , Y/~,

Hence, the group

and the

~ be the normal closure in that belong to ~

~

Then N ~ / ~

and the quotient of y / ~

by this normal subgroup

~-~/~

~/=~/~.~/~

is polycyclic, and

is the

des ired decomposition. In the general case, we covering group ~ = ~ take for

~t

duct ~ / T

first considerthewell-knownimbedding

&/~,

where

~

Let ~ 6 ~

~/~

~

be a torsion-free nilpotent normal subgroup of

It is very well known how one can define a group

which are of the form ~,~ • where ~ E ~ ~ ~ and for the subgroup ~ .

~

in the wreath pro-

{~

~ ,

~ , and

~@

the com-

the elements of

is a fixed set of representatives in

Multiplication is carried out by the rule

where {(~,#)£X , and the action of the element action of

of the image o f ~

We

The properties of ~! are obvious. ,

pletion of ~ .

and the

is the cover of ~ t h a t we have just constructed.

the inverse under the map ~ - - ~ Z .

~--~/~

~

on

~

is the unique extension of the

on N •

LEMMA 4.2.

Let N ..< ~, -.~~

I N I : ~ ~ < ~,

and suppose that

N

is normalized by the

subgroup ~ ; then

(1)

N

(2)

IN:

N C.., N C--/ N

1

Proof.

t

~

=

,I

= 6/N

;

IN,¢ ' C_,l.

The proof is simple and well known.

We will henceforth apply Lemma 4.2 without

special reference. The proof of the following lemma uses an idea of Mal'tsev (cf. [8], and also [9]). LE~gIA 4.3.

7--AM

Let ~ & ~

, ,4 (3 M = ~

and let A be a torsion-free Abelian normal subgroup of ~ .

be a decomposition of some normal subgroup T

Let

of index ra in ~ , where

45

M

is a nilpotent subgroup.

A

in A ~ , such that

~#

then contains a subgroup

Let 4

be an arbitrary

contains the ~ - t h

~

r

in ~ .

roots of elements of

We take a fixed set

~

~--~,~ .... , ~

of coset representatives

A~

for the subgroup

and they are contained in some system of cosets representative is a subgroup

~

, then we put

for the subgroup

of the form ~--~,~,~"I,~,o.,/~

form a system of coset representatives

If the set S

The group ~ -

is an almost-nilpotent group.

Consider the sets of all elements of ~

elements of $

A

of finite index, such that ~ can be represented as ~ - - A ~ ,

with ~ r =- A~M~-- ~, 4 ~ ~ m I , where Proof.

~ -admissible finite extension of

~-

~,

~A~I

.

The

of ~ ( ~

for the subgroup

~

of ~

and the lemma has been proved.

.

If

not, we will alter the elements of ~ , by multiplying them by elements from A , in such a way that

f~

does not change, and the resulting set S ~f is closed under multiplication.

Before establishing the existence of such a procedure, we observe that if we put M|..$1, the lemma is proved.

Indeed, the intersection

in the quotient ~

~/A I

~O~

is trivial, as before.

The image of

is an almost-nilpotent subgroup, from which it follows that

itself is an almost-nilpotent subgroup. To every element

of cardinality

#E $

I~I

we bijectively associate an index

We will write

~--~'=

~p~.

~E~

, where ~ is some set

We have the equality

A. where

~

denotes the corresponding uniquely defined index

is a factor system, associated with ~ . We observe that this case relation

~=~,

and if

~#~=~)~

~'=~,

f. e 7 ,

where

Indeed,

~--~E~

.

in

The

yields the well-known condition (in additive notation):

i.e.,

~EH

means that the function Tin ~ .

~&~,j~= ~ , if ~ M .

then ~ - ~ = ~ - - - - ~ ,

o~,~-), if

IE~ '~. The set {P/IoQ/3);cQDE~}

a,c'j"+,~ - 0~¢~j3,~) +o.~.~)~r

(2)

t h ~ this condition implies that O.Cj3.~')"= O~[~/gg~,~ 3 z ~ ) , ~ = ~ , ~ ) 6 0 ~ .

# has splitting--< 2.

If now only remains to use the polynomials

Along the same lines, one proves the following

48

~= 6 , ~

This proves i).

~ denote the subgroup of the group

We observe that

then [ ~ , ~ ]

We

It then follows from (4) that ~' 8 =- ~

has infinite width relative to the word ~ ~ - ~ . ~ ] , Proof.

(4)

It is obvious that ~ ( ~

If &Cz)

;~, constructed in Lemma 5.1.

Proposition 5.3. T CC ~ )

The group

by the elements

second commutator group Proof of Theorem 2.

~ L~p,~

~ ~

~gg~CO, f ~ ) ~

~

, has the property that the width of the

is unbounded as a function of frg.

The groups

tive to the natural homomorphisms. Let

, generated in the group of triangular matrices

C~,~,~ ~den°te the element

~p~

constructed above, form an inverse system, rela-

The group

~-

~,~(~,~,~) ~ ~

is a homomorphic image of .

The sequence~c~ { C~

n.~.

:ffg~

is an element of [~, which, by Proposition 5.3, does not belong to ~. This proves the theorem. LITERATURE CITED 1.

2. 3. 4. 5. 6. 7. 8.

9.

Kourovskaya Tetrad' [in Russian], 6th ed., Novosibirsk (1968). Yu. I. Merzlyakov, "Algebraic linear groups as full automorphism groups, with closed verbal subgroups," Algebra Logika, 6, No. i, 83-94 (1967). A. H. Rhemtulla, "A problem of bounded expessibility in free products," Proc. Cambridge Phil. Sot., 64, No. 3, 573-584 (1968). A. H. Rhemtulla, "Commutators of certain finitely generated soluble groups," Can. J. Math., 21, No. 5, 1160-1164 (1969). J. Wilson, "On outer-commutator words," Can. J. Math., 26, No. 3, 608-620 (1974). D. I. Zaitsev, "On solvable groups of finite rank," Algebra Logika, 16, No. 3, 300-312 (1977). B. H. Hartley, "Subgroups of finite index in profinite groups," Math. Z., 168, 71-76 (1979). A. I. Mal'tsev, "On some classes of infinite solvable groups," Mat. Sb., 28, No. 3, 567-588 (1951). D. I. Zaitsev, "On groups with normal subgroups that admit complements," Algebra Logika, 14, No. i, 5-14 (1975).

49