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BY 4.0 license Open Access Published by De Gruyter March 20, 2020

On semiconcise words

  • Costantino Delizia , Pavel Shumyatsky and Antonio Tortora
From the journal Journal of Group Theory

Abstract

Let w be a group-word. For a group G, let G w denote the set of all w-values in G and w ( G ) the verbal subgroup of G corresponding to w. The word w is semiconcise if the subgroup [ w ( G ) , G ] is finite whenever G w is finite. The group G is an FC ( w ) -group if the set of conjugates x G w is finite for all x G . We prove that if w is a semiconcise word and G is an FC ( w ) -group, then the subgroup [ w ( G ) , G ] is FC -embedded in G, that is, the intersection C G ( x ) [ w ( G ) , G ] has finite index in [ w ( G ) , G ] for all x G . A similar result holds for BFC ( w ) -groups, that are groups in which the sets x G w are boundedly finite. We also show that this is no longer true if w is not semiconcise.

1 Introduction

Let w = w ( x 1 , , x n ) be a group-word in the variables x 1 , , x n . For any group G and arbitrary g 1 , , g n G , the elements of the form w ( g 1 , , g n ) are called the w-values in G. The set of all w-values in G is denoted by G w . Clearly, any conjugate of a w-value is again a w-value, and so G w is a normal set. The verbal subgroup of G corresponding to w is the subgroup w ( G ) generated by all w-values in G.

A word w is called concise if the verbal subgroup w ( G ) is finite in each group G such that G w is finite. In the sixties, P. Hall conjectured that every word is concise, but his conjecture was refuted in 1989 by S. Ivanov ([8], see also [9]). However, many words of common use are known to be concise (see, e.g., [11, 10, 6, 3]).

Let w be any group-word, and suppose that G w is finite for a group G. It is well known that the derived group w ( G ) is always finite and its order is bounded by a function depending only on the order of G w . In [6], it has been proved that the same is true for the subgroup [ w ( G ) , G ] , when w is the n-Engel word

[ x , n y ] = [ x , y , , y n ] .

Motivated by this, we say that a word w is semiconcise if the finiteness of G w for a group G always implies the finiteness of the subgroup [ w ( G ) , G ] .

Of course, Engel words and concise words are semiconcise. Moreover, if w is a semiconcise word and z is any variable not appearing in w, then the word [ w , z ] is semiconcise (see Proposition 4.2). Thus the word [ x , n y , z ] is semiconcise. So far, we do not know whether there exists a semiconcise word which is not concise. A suitable modification of Ivanov’s example, given in [1], shows that there exists a word which is not semiconcise (see Section 4).

Further information on the subgroup [ w ( G ) , G ] when w is a semiconcise word can be obtained by using a verbal generalization of FC -groups, namely groups with finite conjugacy classes. For subsets X and Y of a group G, we write X Y to denote the set of conjugates { x y x X , y Y } . A subgroup H of a group G is said to be FC -embedded in G if x H is finite for all x G . The subgroup H is BFC -embedded in G if x H is finite for all x G and the number of elements in x H is bounded by a constant that does not depend on the choice of x.

For any group-word w, a group G is said to be an FC ( w ) -group if x G w is finite for all x G , and a BFC ( w ) -group if x G w is finite for all x G and the number of elements in x G w is bounded by a constant that does not depend on the choice of x. Obvious examples of FC ( w ) -groups (respectively, BFC ( w ) -groups) are provided by groups G in which the verbal subgroup w ( G ) is FC -embedded (respectively, BFC -embedded) in G. On the other side, it has been proved in [7] (respectively, in [1]) that if w is a concise word and G is an FC ( w ) -group (respectively, a BFC ( w ) -group), then the verbal subgroup w ( G ) is FC -embedded (respectively, BFC -embedded) in G.

In [2], for an arbitrary group-word w, we proved that if G is an FC ( w ) -group (respectively, a BFC -group), then w ( G ) is FC -embedded (respectively, BFC -embedded) in G. In this paper, we restrict our attention to semiconcise words. More precisely, we prove the following theorems.

Theorem A.

Let w be a semiconcise word, and let G be an FC ( w ) -group. Then [ w ( G ) , G ] is FC -embedded in G.

Theorem B.

Let w be a semiconcise word, and let G be a BFC ( w ) -group. Then [ w ( G ) , G ] is BFC -embedded in G.

We also show that, for a certain word w which is not semiconcise, there is an example of a BFC ( w ) -group G such that [ w ( G ) , G ] is not FC -embedded in G (see Proposition 4.4).

2 Proof of Theorem A

For a subset S of a group G, write S * = S S - 1 . Obviously, if S is a normal set, then S * is normal too. Moreover, if S is a finite set, we often refer to the “order of S”, denoted by | S | , to mean “the number of elements in S”.

Lemma 2.1.

Let w be a group-word. If G is an FC ( w ) -group, then the conjugacy class x G w * is finite for all x G .

Proof.

By the first statement in [2, Proposition 2.9], a group G is an FC ( w ) -group if and only if it is an FC ( w - 1 ) -group. For all g G , we have g G w if and only if g - 1 G w - 1 . Thus G w - 1 = ( G w ) - 1 , and the result follows. ∎

Lemma 2.2.

Let w be a group-word, and let G be an FC ( w ) -group. Choose x G , and denote by A a finite subset of G w * such that x G w * = x A . Then, for any j 1 and y 1 , , y j G w * , there exist a 1 , , a j A such that x y 1 y j = x a 1 a j .

Proof.

We argue by induction on j. The case j = 1 is clear. Let j > 1 , and assume that x y 1 y j - 1 = x a 1 a j - 1 with a 1 , , a j - 1 A . Then

x y 1 y j = x a 1 a j - 1 y j = x y j b a 1 a j - 1 ,

where b = ( a 1 a j - 1 ) - 1 . Since y j b G w * , we have x y j b = x a j for some a j A , and so x y 1 y j = x a j a 1 a j - 1 . After renumbering the elements a i A , we obtain the required result. ∎

Lemma 2.3.

Let w = w ( x 1 , , x n ) be a group-word, and set

v = [ w ( x 1 , , x n ) , x n + 1 ] .

If G is an FC ( w ) -group, then it is an FC ( v ) -group.

Proof.

Let y G v . Then there exist g 1 , , g n , g n + 1 G such that y = z t , with z = w ( g 1 , , g n ) - 1 ( G w ) - 1 = G w - 1 and t = w ( g 1 , , g n ) g n + 1 G w . By Lemma 2.2, for any x G , the conjugate x y can only take finitely many values. This proves the result. ∎

Lemma 2.4.

Let w = w ( x 1 , , x n ) be a semiconcise word, and set

v = [ w ( x 1 , , x n ) , x n + 1 ] .

Let G be an FC ( w ) -group and B a finite subset of G v * . Then, for any x G , there exists a positive integer e such that b e Z ( x , B ) for all b B .

Proof.

Write B = { b 1 , , b r } , and let x be an arbitrary element of G. For any b i B , there exist elements g i 1 , , g i n , g i n + 1 G such that either

b i = [ w ( g i 1 , , g i n ) , g i n + 1 ] or b i = [ w ( g i 1 , , g i n ) , g i n + 1 ] - 1 .

Put

J = x , g i j 1 i r , 1 j n + 1 .

By [2, Lemma 2.7 (i)], the set ( J / Z ( J ) ) w is finite. As w is semiconcise, the subgroup [ w ( J / Z ( J ) ) , J / Z ( J ) ] is finite. Thus v ( J ) has finite order modulo Z ( J ) , say e. Since B v ( J ) , it follows that b i e Z ( J ) for all i. As x , B J , the result follows. ∎

Proof of Theorem A.

Set v = [ w ( x 1 , , x n ) , x n + 1 ] . Then G is an FC ( v ) -group by Lemma 2.3. Let x be an arbitrary element of G. By Lemma 2.1, we can choose b 1 , , b r G v * such that x G v * = { x b 1 , , x b r } . Write B = { b 1 , , b r } . Define the order < on the set of all (formal) products of the form b i 1 b i j , with 1 i k r and j 1 , as follows. Put

(2.1) b i 1 b i j < b i 1 b i j

if and only if one of the following conditions is satisfied: j < j , or j = j and there is a positive integer l j such that i l < i l and i k = i k for all k > l .

Let y be an arbitrary element of v ( G ) . Then y = y 1 y j , where each y i G v * . By Lemma 2.2, for all k { 1 , , j } , there exists an integer i k { 1 , , r } such that x y = x b i 1 b i j . Clearly, we can choose b i 1 b i j to be the smallest (in the sense of the order < ) product of elements from B such that x y = x b i 1 b i j . Let us now show that i 1 i 2 i j . Suppose to the contrary that i k < i k + 1 for some k. Then

x y = x b i 1 b i k - 1 b i k b i k + 1 b i k + 2 b i j = x b i 1 b i k - 1 c b i k b i k + 2 b i j ,

where c = b i k b i k + 1 b i k - 1 G v * . In view of Lemma 2.2, we have

x b i 1 b i k - 1 c = x b i 1 b i k - 1 b i k + 1

for some 1 i 1 , , i k - 1 , i k + 1 r so that

x y = x b i 1 b i k - 1 b i k + 1 b i k b i k + 2 b i j .

This contradicts the choice of the product b i 1 b i j because

b i 1 b i k - 1 b i k b i k + 1 b i k + 2 b i j > b i 1 b i k - 1 b i k + 1 b i k b i k + 2 b i j .

Thus x y = x b i 1 b i j with i 1 i 2 i j or, equivalently,

x y = x b r e r b 1 e 1

for some non-negative integers e r , , e 1 .

By Lemma 2.4, there exists a positive integer e such that b i e Z ( x , B ) for all i. Hence we may assume that e i < e for all i, and so | x v ( G ) | e r . Thus x v ( G ) is finite for all x G . We conclude therefore that v ( G ) = [ w ( G ) , G ] is FC -embedded in G, as required. ∎

3 Proof of Theorem B

The proof of Theorem B is very similar to that of Theorem A; the only important difference is the presence of bounds. In what follows, the term “ { a , b , c , } -bounded” means “bounded from above by some function depending only on the parameters a , b , c , ”.

Before proving Theorem B, we recall briefly the ultraproduct construction of groups (see, for instance, [4] for more details).

For a non-empty set I, a filter over I is a set with the following properties:

  1. , I ;

  2. if X , Y , then X Y ;

  3. if X and X Y I , then Y .

The filter is principal if there exists a non-empty set Y I such that

= { X I Y X } ,

and non-principal otherwise. An example of a non-principal filter over an (infinite) set I is the so-called cofinite filter

= { X I I X is finite } .

A filter 𝒰 over I is called an ultrafilter if, for every X I , either X 𝒰 or I X 𝒰 . This is equivalent to saying that 𝒰 is a maximal filter over I. Also, 𝒰 is a non-principal ultrafilter if and only if it contains the cofinite filter (see [4, Proposition 1.4]).

Given an ultrafilter 𝒰 over I and a family { G i } i I of groups, the ultraproduct modulo 𝒰 is the quotient set of the Cartesian product i I G i with respect to the equivalence relation defined as follows: the tuples ( g i ) i I and ( h i ) i I of the Cartesian product are equivalent modulo 𝒰 if and only if

{ i I g i = h i } 𝒰 .

Thus the ultraproduct modulo 𝒰 can be seen as the quotient of the unrestricted direct product of groups G i by the subgroup consisting of all tuples ( g i ) i I such that

{ i I g i = 1 } 𝒰 .

The following is a consequence of Łoś’s theorem (see [4, Corollary 3.2]), in the case of a sentence in the first-order language of groups.

Lemma 3.1.

Let { G i } i I be a family of groups, and let U be an ultrafilter over I. Then a sentence in the first-order language of groups holds in the ultraproduct modulo U if and only if the set of all i I for which the sentence holds in G i is a member of U .

Recall that the width of a group-word w in a group G is the supremum, as g ranges over the verbal subgroup w ( G ) , of the minimum length of all decompositions of g as a product of elements of G w * . Clearly, for a given positive integer k, the word w has finite width at most k in G if and only if the product of any k + 1 elements of G w * can be expressed as a product of at most k elements of G w * . Our next result relies on Lemma 3.1 and the following two facts, which have been stated in [5, proof of Theorem A.1]:

  1. for a given integer m, the property that a given word takes at most m values in a group can be expressed as a sentence in the first-order language of groups;

  2. for a given positive integer k, the property that a given word has finite width at most k in a group can be expressed as a sentence in the first-order language of groups.

Proposition 3.2.

Let m 1 . Suppose that w is a semiconcise word and G is a group in which w takes precisely m values. Then the order of [ w ( G ) , G ] is m-bounded.

Proof.

Assuming that w involves n variables, write w = w ( x 1 , , x n ) , and set v ( x 1 , , x n , x n + 1 ) = [ w ( x 1 , , x n ) , x n + 1 ] . Then [ w ( G ) , G ] = v ( G ) .

By way of contradiction, suppose there exists a family of groups 𝒢 = { G i } i with the property that | ( G i ) w | m for all i but

lim i | v ( G i ) | = lim i | [ w ( G i ) , G i ] | = .

Consider a non-principal ultrafilter 𝒰 over , and let Q be the ultraproduct modulo 𝒰 of 𝒢 . Then, by (a) and Lemma 3.1, we have | Q w | m . As w is semiconcise, it follows that v ( Q ) = [ w ( Q ) , Q ] is finite. In particular, v has finite width, say k, in Q. Now (b) and Lemma 3.1 yield that there exists X 𝒰 such that v has finite width at most k in G i for all i X . Hence every element of v ( G i ) = [ w ( G i ) , G i ] can be written as a product of at most k elements of ( G i ) v * . Moreover, as

v ( x 1 , , x n , x n + 1 ) = w ( x 1 , , x n ) - 1 w ( x 1 , , x n ) x n + 1 ,

from | ( G i ) w | m , we get | ( G i ) v | m 2 for all i . Therefore, | v ( G i ) | m 2 k for all i X . As noted above, 𝒰 contains the cofinite filter over , so X Y 𝒰 for every cofinite subset Y of . In particular, X Y is non-empty. Therefore, every cofinite subset of contains some element i for which | v ( G i ) | m 2 k . This is incompatible with the assumption that | v ( G i ) | goes to infinity. ∎

Lemma 3.3.

Let w = w ( x 1 , , x n ) be a group-word. If G is a BFC ( w ) -group such that | x G w | m for all x G , then the conjugacy class x G w * has { m , n } -bounded order for all x G .

Proof.

By the second statement in [2, Proposition 2.9], a group G is a BFC ( w ) -group if and only if it is a BFC ( w - 1 ) -group. More precisely, if G is a BFC ( w ) -group such that | x G w | m for all x G , then x G w - 1 has { m , n } -bounded order. Since G w * = G w G w - 1 , the result follows. ∎

Lemma 3.4.

Let w = w ( x 1 , , x n ) be a group-word, and set

v = [ w ( x 1 , , x n ) , x n + 1 ] .

If G is a BFC ( w ) -group such that | x G w | m for all x G , then G is a BFC ( v ) -group and x G v has { m , n } -bounded order for all x G .

Proof.

This is similar to the proof of Lemma 2.3 taking into account that x y can only take { m , n } -boundedly many values by Lemma 3.3. ∎

Lemma 3.5.

Let w = w ( x 1 , , x n ) be a group-word, and set

v = [ w ( x 1 , , x n ) , x n + 1 ] .

Let G be a BFC ( w ) -group such that | x G w | m for all x G , and let B be a finite subset of G v * . Then, for any x G , there exists an { m , n , | B | } -bounded positive integer e such that b e Z ( x , B ) for all b B .

Proof.

It follows as in the proof of Lemma 2.4: by [2, Lemma 2.7 (ii)], the set ( J / Z ( J ) ) w is finite of { m , n , | B | } -bounded order and, by Proposition 3.2, the number e is { m , n , | B | } -bounded. ∎

Proof of Theorem B.

Set v = [ w ( x 1 , , x n ) , x n + 1 ] . Then Lemma 3.4 tells us that G is a BFC ( v ) -group and x G v has { m , n } -bounded order for all x G . Let x be an arbitrary element of G, and choose

b 1 , , b r G v * such that x G v * = { x b 1 , , x b r } .

Write B = { b 1 , , b r } . Define the order < on the set of all (formal) products of the form b i 1 b i j , with 1 i k r and j 1 , as in (2.1) in the proof of Theorem A.

Let y be an arbitrary element of v ( G ) . Arguing as in the proof of Theorem A, write

x y = x b r e r b 1 e 1

for some non-negative integers e r , , e 1 . Since the number r is { m , n } -bounded by Lemma 3.4, it follows from Lemma 3.5 that there exists an { m , n } -bounded positive integer e such that b i e Z ( x , B ) for all i. Hence we may assume that e i < e for all i, and so | x v ( G ) | e r . Thus x v ( G ) is finite of { m , n } -bounded order for all x G , and so v ( G ) is BFC -embedded in G. ∎

4 Examples

In this section, we first provide some new examples of semiconcise words, and then we give an example of a word which is not semiconcise.

The following lemma is well known (see, for instance, [10, Lemma 4.28]).

Lemma 4.1.

Let G = g 1 , , g m be a group, and suppose that, for a group-word w, the set { [ x , g i ] x G w , i = 1 , , m } is finite. Then G w is contained in finitely many right cosets of w ( G ) Z ( G ) .

Proof.

Suppose that there exists an infinite sequence x 1 , x 2 , of elements of G w belonging to distinct right cosets of C = w ( G ) Z ( G ) . For any i = 1 , , m , put S i = { [ x , g i ] x G w } . Denote by P the Cartesian product S 1 × × S m , and let π be the map sending x G w to ( [ x , g 1 ] , , [ x , g m ] ) P . Now we have that x i π = x j π implies x i x j - 1 C , or C x i = C x j which is impossible. It follows that P must be infinite, a contradiction. ∎

Proposition 4.2.

Let w = w ( x 1 , , x n ) be a group-word, and set

v = [ w ( x 1 , , x n ) , x n + 1 ] .

If w is semiconcise, then v is semiconcise.

Proof.

Let G be a group, and assume that G v is finite. Since v ( G ) is finite (see, for instance, [6, Proposition 1]), we may assume that v ( G ) is abelian. It follows that every subgroup of v ( G ) is finitely generated. Let K = g 1 , , g m be a finitely generated subgroup of G such that v ( G ) = v ( K ) . For an arbitrary g = g 0 G , put H = g 0 , K . Of course, v ( G ) = v ( H ) . Also, | H : C H ( v ( H ) ) | is finite because every h H v has only finitely many conjugates in H. We claim that | v ( H ) : v ( H ) Z ( H ) | is also finite, from which it follows that [ v ( H ) , H ] is finite by a result of Baer (see [10, Corollary, p. 103]). In fact, the set

{ [ x , g i ] x H w , i = 0 , 1 , , m }

is finite, and therefore, by Lemma 4.1, H w is contained in finitely many right cosets of w ( H ) Z ( H ) . Hence ( H / Z ( H ) ) w is finite. Since w is semiconcise, we obtain that

[ w ( H ) , H ] Z ( H ) / Z ( H ) v ( H ) / v ( H ) Z ( H )

is finite.

Then [ v ( H ) , H ] = [ v ( G ) , H ] is finite. In particular, [ v ( G ) , g ] is finite for any g G . Thus [ v ( G ) , G ] is a finitely generated periodic abelian group, and so [ v ( G ) , G ] is finite. This proves that v is semiconcise. ∎

As an immediate consequence of Proposition 4.2, we get new examples of semiconcise words starting from Engel words, which are semiconcise by [6, Proposition 4].

Corollary 4.3.

Let w = [ x , n y ] be the n-Engel word, and set

v = [ x , n y , z 1 , , z m ] ,

where the variables x , y , z 1 , , z m are all different. Then v is semiconcise.

According to [8], for any odd integer n > 10 10 and any prime number p > 5000 , the word

v ( x , y ) = [ [ x p n , y p n ] n , y p n ] n

is not concise. Indeed, Ivanov constructed a 2-generator torsion-free group A, whose center is cyclic and A / Z ( A ) is infinite of exponent p 2 n , such that v takes only two values in A and the nontrivial value is a generator of Z ( A ) .

In [1, Section 4], the authors considered a modification of Ivanov’s example, namely the wreath product

(4.1) G = wr A B ,

where A is as above and B = b is a cyclic group of order 2. Furthermore, taking

(4.2) w ( x , y ) = v ( x 2 , y 2 ) ,

they showed that | G w | 4 and b w ( G ) is infinite. In a similar way, we now prove that b [ w ( G ) , G ] is also infinite. This implies that w is not semiconcise.

Proposition 4.4.

There exist a group-word w (which is not semiconcise) and a BFC ( w ) -group G such that [ w ( G ) , G ] is not FC -embedded in G.

Proof.

Let G and w be as in (4.1) and (4.2), respectively. Then, by [1, Proposition 4.1], G is a BFC ( w ) -group.

Denote by K = A × A b the base group of G. For any odd integer m 1 , let N = v 0 m , ( v 0 b ) m , where v 0 A is the nontrivial value of v ( x , y ) in A. Notice that N is central in K and closed under conjugation by b so that N is a normal subgroup of G. Also, since K / N has odd exponent p 2 m n and | G / K | = 2 , we have

K / N = { g 2 N g G }

and consequently

( K / N ) v = ( G / N ) w .

Hence v 0 N ( G / N ) w , and therefore v 0 k N w ( G / N ) for any integer k. It follows that

b [ b , v 0 k ] N ( b N ) [ w ( G / N ) , G / N ] .

Now

b [ b , v 0 k ] = b [ b , v 0 k , b ] - 1 = b [ ( v 0 b ) - k v 0 k , b ] - 1 = b ( v 0 2 ( v 0 b ) - 2 ) k ,

where v 0 2 ( v 0 b ) - 2 has order m in G / N . Thus

| { b [ b , v 0 k ] N k } | = m ,

and so

| ( b N ) [ w ( G / N ) , G / N ] | m .

In particular, | b [ w ( G ) , G ] | m . Since m is an arbitrary odd positive integer, we conclude that b [ w ( G ) , G ) ] is infinite. This proves that [ w ( G ) , G ) ] is not FC ( w ) -embedded in G. ∎


Communicated by Evgenii I. Khukhro


Funding statement: This work was partially supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA – INdAM). The last author is also supported by a grant of the University of Campania “Luigi Vanvitelli”, in the framework of Programma V:ALERE 2019.

Acknowledgements

The authors would like to thank the referee for a number of useful suggestions and, in particular, for correcting a mistake in an earlier version of the proof of Proposition 3.2.

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Received: 2019-07-24
Revised: 2020-02-07
Published Online: 2020-03-20
Published in Print: 2020-07-01

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