Bounded conjugacy classes, commutators, and approximate subgroups
aa r X i v : . [ m a t h . G R ] F e b Bounded conjugacy classes, commutators,and approximate subgroups
Pavel Shumyatsky
Abstract.
Given a group G , we write g G for the conjugacy classof G containing the element g . A famous theorem of B. H. Neu-mann states that if G is a group in which all conjugacy classesare finite with bounded size, then the commutator subgroup G ′ isfinite. We establish the following results.Let K, n be positive integers and G a group having a K -approximatesubgroup A .If | a G | ≤ n for each a ∈ A , then the commutator subgroup of h A G i has finite ( K, n )-bounded order.If | [ g, a ] G | ≤ n for all g ∈ G and a ∈ A , then the commutatorsubgroup of [ G, A ] has finite (
K, n )-bounded order.
1. Introduction
Let K be a positive integer. A symmetric subset A of a group G is a K -approximate subgroup if there is a finite subset E ⊆ G suchthat | E | ≤ K and AA ⊆ EA . The formal definition of an approximatesubgroup was introduced by Tao in [ ]. Since then many importantresults on approximate subgroups have been established. In particular,Breuillard, Green and Tao essentially described the structure of finiteapproximate subgroups [ ]. The reader is referred to the recent book[ ], or the surveys [
2, 3 ], for detailed information on these develop-ments.An interesting principle was stated in [ ]: Mathematics Subject Classification.
Key words and phrases.
Conjugacy classes, commutators, approximatesubgroups.This research was supported by the Conselho Nacional de DesenvolvimentoCient´ıfico e Tecnol´ogico (CNPq), and Funda¸c˜ao de Apoio `a Pesquisa do DistritoFederal (FAPDF), Brazil.
Group-theoretical arguments can often be successfully transferred toapproximate subgroups .In the present article we check this principle against certain varia-tions of B. H. Neumann’s theorem that a BFC-group has finite com-mutator subgroup.Given a group G and an element x ∈ G , we write x G for the conju-gacy class containing x . More generally, if X, Y ⊆ G , we write X Y forthe set of all x y , where x ∈ X and y ∈ Y . Of course, if the number ofelements in x G is finite, we have | x G | = [ G : C G ( x )]. A group is calleda BFC-group if its conjugacy classes are finite and have bounded size.In 1954 B. H. Neumann discovered that the commutator subgroup G ′ of a BFC-group G is finite [ ]. It follows that if | x G | ≤ n for each x ∈ G , then G ′ has finite n -bounded order. Throughout the articlewe use the expression “( a, b, . . . )-bounded” to mean that a quantity isfinite and bounded by a certain number depending only on the param-eters a, b, . . . . A first explicit bound for the order of G ′ was found by J.Wiegold [ ], and the best known was obtained in [ ] (see also [ ] and[ ]). The article [ ] deals with groups G in which conjugacy classescontaining commutators are bounded. It is shown that if | x G | ≤ n forany commutator x , then the second commutator subgroup G ′′ has finite n -bounded order. Later this was extended in [ ] to higher commutatorsubgroups. A related result for groups in which the conjugacy classescontaining squares have finite bounded sizes was obtained in [ ].A stronger version of the Neumann theorem was recently estab-lished in [ ]: Let n be a positive integer and G a group having a subgroup A suchthat | a G | ≤ n for each a ∈ A . Then the commutator subgroup of h A G i has finite n -bounded order. Here, as usual, h X i denotes the subgroup generated by the set X and so h A G i denotes the minimal normal subgroup containing A .In the present paper we extend the above result as follows. Theorem . Let
K, n be positive integers and G a group havinga K -approximate subgroup A such that | a G | ≤ n for each a ∈ A . Thenthe commutator subgroup of h A G i has finite ( K, n ) -bounded order. For a subset X of a group G we write [ G, X ] to denote the subgroupgenerated by all commutators [ g, x ], where g ∈ G and x ∈ X . It iswell-known that [ G, X ] is a normal subgroup of G . Moreover, [ G, X ] =[ G, h X i ]. We examine approximate subgroups A ⊆ G such that the FC-THEOREMS FOR APPROXIMATE SUBGROUPS 3 conjugacy classes of commutators [ g, a ] have bounded sizes whenever g ∈ G and a ∈ A . Theorem . Let
K, n be positive integers and G a group havinga K -approximate subgroup A such that | [ g, a ] G | ≤ n for all g ∈ G and a ∈ A . Then the commutator subgroup of [ G, A ] has finite ( K, n ) -bounded order. It is worthwhile to mention that Theorem 1.2 was unknown evenin the case where A is a subgroup of G . It can be regarded as anextension of the aforementioned result in [ ] that if | x G | ≤ n for anycommutator x , then the second commutator subgroup G ′′ has finite n -bounded order.
2. Preliminaries
Let G be a group generated by a set X such that X = X − . Givenan element g ∈ G , we write l X ( g ) for the minimal number l with theproperty that g can be written as a product of l elements of X . A proofof the following result can be found in [ , Lemma 2.1]. Lemma . Let G be a group generated by a set X = X − and let L be a subgroup of finite index m in G . Then each coset Lb containsan element g such that l X ( g ) ≤ m − . The next lemma is almost obvious.
Lemma . Let k, n, s ≥ , and let G be a group containing a set X = X − such that | x G | ≤ n for any x ∈ X . Let g , . . . , g s ∈ h X i and assume that l X ( g i ) ≤ k . Then C G ( g , . . . , g s ) has finite ( k, n, s ) -bounded index in G . Proof.
Since l X ( g i ) ≤ k , we can write g i = x i . . . x ik , where x ij ∈ X and i = 1 , . . . , s . By the hypothesis the index of C G ( x ij ) in G isat most n for any such element x ij . Set U = ∩ i,j C G ( x ij ). We have[ G : U ] ≤ n ks . Since U ≤ C G ( g , . . . , g s ), the lemma follows. (cid:3) The next observation will play a crucial role in the proof of Theo-rems 1.1 and 1.2.
Lemma . Let X be a normal subset of a group G such that | x G | ≤ n for any x ∈ X , and let H = h X i . Then the subgroup h [ H, x ] G i has finite n -bounded order. Proof.
Without loss of generality we can assume that X = X − .Let m be the maximum of indices of C H ( x ) in H for x ∈ X . Of course, m ≤ n . Take x ∈ X . Since the index of C H ( x ) in H is at most PAVEL SHUMYATSKY m , by Lemma 2.1 we can choose elements y , . . . , y m in H such that l X ( y i ) ≤ m − H, x ] is generated by the commutators[ y i , x ], for i = 1 , . . . , m . For any such i write y i = y i . . . y i ( m − , with y ij ∈ X . By using standard commutator identities we can rewrite[ y i , x ] as a product of conjugates in H of the commutators [ y ij , x ].Let { h , . . . , h s } be the conjugates in H of all elements from the set { x, y ij | ≤ i ≤ m, ≤ j ≤ m − } . Note that the number s here is m -bounded. This follows form the fact that C H ( x ) has index at most m in H for each x ∈ X . Put T = h h , . . . , h s i . Observe that the centre Z ( T ) has index at most m s in T , since the index of C H ( h i ) in H is atmost m for any i = 1 , . . . , s . Thus, by Schur’s theorem [ , 10.1.4],we conclude that the commutator subgroup T ′ has finite m -boundedorder. Since [ H, x ] is contained in T ′ , deduce that the order of [ H, x ] is m -bounded. Further, the subgroup [ H, x ] is normal in H and there areat most n conjugates of [ H, x ] in G . Therefore h [ H, x ] G i is a productof at most n normal subgroups, each of which has n -bounded order.Hence, the result. (cid:3) Lemma . Let G be a group and x, y ∈ G . Assume that | x G | = m and | ( yx ) G | ≤ m . Suppose that there are b , . . . , b m ∈ G such that x G = { x b , . . . , x b m } and y ∈ C G ( b , . . . , b m ) . Then [ G, y ] ≤ [ G, x ] . Proof.
First we note that ( yx ) G = { yx b , . . . , yx b m } . This is be-cause all elements yx b , . . . , yx b m are different and since by the hy-pothesis ( yx ) G contains at most m elements, the class ( yx ) G mustcoinside with { yx b , . . . , yx b m } . Therefore for any g ∈ G there is b i ∈ { b , . . . , b m } such that ( yx ) g = yx b i . So y g x g = yx b i and [ y, g ] = x b i x − g ∈ [ G, x ]. The lemma follows. (cid:3)
3. Proof of Theorem 1.1
Recall that A is a K -approximate subgroup of a group G such that | a G | ≤ n for any a ∈ A . We wish to show that H = h A G i has finitecommutator subgroup of n -bounded order.Let E = { e , . . . , e K } be a set of size K such that AA ⊆ EA .It will be assumed that K is chosen as small as possible and so foreach i = 1 , . . . , K there are x i , x i , x i ∈ A such that x i x i = e i x i .By Lemma 2.3 each of the subgroups h [ H, x ij ] G i has n -bounded order.Hence, also their product has finite ( K, n )-bounded order. Now we canpass to the quotient over the product of all h [ H, x ij ] G i and without lossof generality assume that the set E is contained in the centre of H .Denote by X the set A G . Let m be the maximum of indices of C H ( x ) in H for x ∈ X . Of course, m ≤ n . Select a ∈ A such that | a H | = m . Choose b , . . . , b m in H such that l X ( b i ) ≤ m − FC-THEOREMS FOR APPROXIMATE SUBGROUPS 5 a H = { a b i ; i = 1 , . . . , m } . The existence of the elements b i is guaranteedby Lemma 2.1. Set U = C G ( h b , . . . , b m i ). In view of Lemma 2.2 notethat the index of U in G is n -bounded.Let r be the minimal number for which there are elements d , . . . , d r ∈ A such that A is contained in the union of the left cosets d i U . We fixthe elements d i and denote by S the product of the subgroups h [ H, d i ] G i for i = 1 , . . . , r . By Lemma 2.3 S is a product of at most r normalsubgroups of finite n -bounded order. Taking into account that r is n -bounded conclude that S has finite n -bounded order.Choose any u ∈ A ∩ U . Since AA ⊆ EA write ua = ex for suitable e ∈ E and x ∈ A . Since e ∈ Z ( H ) and | x H | ≤ m , it follows that | ( ua ) H | ≤ m . Recall that U = C G ( h b , . . . , b m i ). Lemma 2.4 impliesthat [ H, u ] ≤ [ H, a ]. This happens for every choice of u ∈ A ∩ U andso [ H, ( A ∩ U )] ≤ [ H, a ].Let T = h [ H, a ] G i and observe that by virtue of Lemma 2.3 T hasfinite n -bounded order. Choose an arbitrary element d ∈ A . There isan index j such that d ∈ d j U and d j − d ∈ U . Since d j − d ∈ AA , write d j − d = ey for suitable e ∈ E and y ∈ A . Observe that e ∈ U , whence y ∈ U . Taking into account that [ H, d j ] ≤ S , [ H, e ] = 1, and [
H, y ] ≤ T deduce [ H, d ] = [
H, d j ey ] ≤ [ H, d j ][ H, e ][ H, y ] ≤ ST.
Since d was chosen in A arbitrarily, conclude that [ H, A ] ≤ ST .Recall that H = h A G i . We therefore conclude that H ′ ≤ ST . Now thetheorem follows from the fact that the order of ST is n -bounded. Thiscompletes the proof of the theorem.
4. Proof of Theorem 1.2
Throughout this section A ⊆ G is a K -approximate subgroup of agroup G such that | [ g, a ] G | ≤ n for each g ∈ G and a ∈ A . We need toprove that [ G, A ] has finite commutator subgroup of (
K, n )-boundedorder.Let X be the set of all conjugates of commutators [ g, a ], where g ∈ G and a ∈ A . Note that the set X is symmetric. Put H = h X i . ByLemma 2.3 the subgroup h [ H, x ] G i has finite n -bounded order whenever x ∈ X . Lemma . For any a ∈ A the subgroup [ H, [ G, a ]] has finite n -bounded order. Proof.
Choose a ∈ A . Let m be the maximum of indices of C H ( x ) in H , where x ranges through the set of commutators [ g, a ] with g ∈ G . Select g ∈ G such that | [ g , a ] H | = m . Choose b , . . . , b m in PAVEL SHUMYATSKY H such that l X ( b i ) ≤ m − g , a ] H = { [ g , a ] b i ; i = 1 , . . . , m } .(The existence of the elements b i is guaranteed by Lemma 2.1.) Set U = C G ( h b , . . . , b m i ). Note that by Lemma 2.2 the index of U in G is n -bounded. Let U = ∩ g ∈ G U g be the maximal normal subgroup of G contained in U . Obviously, the index of U in G is n -bounded as well.For any g ∈ G observe that [ gg , a ] = [ g, a ] g [ g , a ]. Choose g ∈ U andset [ g, a ] g = u . Lemma 2.4 shows that [ H, u ] ≤ [ H, [ g , a ]].Let c , . . . , c k be a transversal of U in G . For i = 1 , . . . , k let T i denote the subgroup h [ H, [ c i , a ]] G i . In view of Lemma 2.3 each subgroup T i has finite n -bounded order. Further, let T denote the subgroup h [ H, [ g , a ]] G i . Likewise, T has finite n -bounded order. Let N be theproduct of all T i for i = 0 , , . . . , k .Any element g ∈ G can be written as a product g = xc j for suitable x ∈ U and j ≤ k . Then we have [ g, a ] = [ xc j , a ] = [ x, a ] c j [ c j , a ]. Wenow know that the images in G/N of both [ x, a ] and [ c j , a ] are centralin H/N . It follows that also the image [
G, a ] is central in
H/N . Since N has finite n -bounded order, the lemma follows. (cid:3) Let E = { e , . . . , e K } be a set of size K such that AA ⊆ EA . Itwill be assumed that K is chosen as small as possible and so for each i = 1 , . . . , K there are x i , x i , x i ∈ A such that x i x i = e i x i . ByLemma 4.1 for each x ij the subgroup N ij = [ H, [ G, x ij ]] has finite n -bounded order. Let N be the product of all these subgroups N ij andobserve that N has finite ( K, n )-bounded order. Pass to the quotient
G/N and assume that [ H, [ G, x ij ]] = 1 for all i, j . Then of course[ H, [ G, e i ]] = 1 for all i = 1 , . . . , K . Therefore in what follows, withoutloss of generality, we will assume that[ G, E ] ≤ Z ( H ) . Let m be the maximum of indices of C H ( x ) in H , where x rangesthrough the set X . Lemma . For any b ∈ h A i and g ∈ G we have | [ g, b ] H | ≤ m . Proof.
Indeed, since b ∈ h A i , we can write b = ea for suitable a ∈ A and e ∈ h E i . Then [ g, b ] = [ g, ea ] ∈ [ G, E ] X . Now use that[ G, E ] ≤ Z ( H ) and | x H | ≤ m for any x ∈ X and deduce the lemma. (cid:3) Now fix h ∈ G and a ∈ A such that | [ h , a ] H | = m . Choose h , . . . , h m in H such that l X ( h i ) ≤ m − h , a ] H = { [ h , a ] h i ; i = 1 , . . . , m } . The existence of the elements h i follows from Lemma 2.1. Set V = C G ( h h , . . . , h m i ). Note that by Lemma 2.2 the index of V in G is FC-THEOREMS FOR APPROXIMATE SUBGROUPS 7 n -bounded. Let V = ∩ g ∈ G V g be the maximal normal subgroup of G contained in V and note that the index of V in G is n -bounded aswell. By Lemma 4.1 the subgroup S = [[ H, [ G, a ]] has finite n -boundedorder. Lemma . [ H, [ G, V ∩ h A i ]] ≤ S . Proof.
Let b ∈ V ∩ h A i . Lemma 4.2 tells us that | [ g, a b ] H | ≤ m for any g ∈ G . Moreover, observe that [ g, a b ] = [ g, b ][ g, a ] b while[ g, b ] ∈ V . Lemma 2.4 shows that [ H, [ g, b ]] ≤ [ H, [ g, a ]] ≤ S . Thishappens for every g ∈ G so the lemma follows. (cid:3) Let r be the minimal number for which there are elements d , . . . , d r ∈ A such that A is contained in the union of the left cosets d V , . . . , d r V .We fix the elements d i and for each i = 1 , . . . , r put M i = [ H, [ G, d i ]].By Lemma 4.1 the product of all M i has finite n -bounded order. Passto the quotient G/ Q i M i and, without loss of generality, assume that[ G, d i ] ≤ Z ( H ) for each i .For an arbitrary element a ∈ A there is i ≤ r such that a ∈ d i V .We have [ G, a ] ≤ [ G, d i ][ G, d i − a ]. By assumptions, [ H, [ G, d i ]] = 1.Taking into account that d i − a ∈ V ∩ h A i and using Lemma 4.3 deducethat [ H, [ G, d i − a ]] ≤ S . Thus, [ H, [ G, a ]] ≤ S whenever a ∈ A . Since H = Q a ∈ A [ G, a ], it follows that H ′ ≤ S . This completes the proof ofthe theorem. References , Cambridge Univ.Press, Cambridge, 2014, pp. 23–50.[4] E. Breuillard, B. Green and T. Tao, The structure of approximate groups,Publ. Math. IHES, (2012), 115–221.[5] E. Detomi, M. Morigi, P. Shumyatsky, BFC-theorems for higher commutatorsubgroups, Quarterly J. Math. (2019), no. 3, 849–858.[6] G. Dierings, P. Shumyatsky, Groups with boundedly finite conjugacy classesof commutators, Quarterly J. Math. (2018), no. 3, 1047–1051.[7] G. Dierings, P. Shumyatsky, Groups in which squares have boundedly manyconjugates, J. Group Theory (2019), 133–136.[8] R. M. Guralnick, A. Maroti, Average dimension of fixed point spaces withapplications, J. Algebra (2011), 298–308.[9] B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc.(3) (1954), 236–248. PAVEL SHUMYATSKY [10] P. M. Neumann, M.R. Vaughan-Lee, An essay on BFC groups, Proc. Lond.Math. Soc. (1977), 213–237.[11] D. J. S. Robinson, A course in the theory of groups , Second edition. GraduateTexts in Mathematics, 80. Springer-Verlag, New York, 1996.[12] D. Segal, A. Shalev, On groups with bounded conjugacy classes, Quart. J.Math. Oxford (1999), 505–516.[13] T. Tao, Product set estimates for non-commutative groups, Combinatorica, (2008), 547–594.[14] M. C. H. Tointon, Introduction to approximate groups, Cambridge UniversityPress, Cambridge, 2020.[15] J. Wiegold, Groups with boundedly finite classes of conjugate elements, Proc.Roy. Soc. London Ser. A (1957), 389–401. Pavel Shumyatsky: Department of Mathematics, University of Brasilia,Brasilia-DF, 70910-900 Brazil
Email address ::