Abstract In 1954, B.H. Neumann discovered that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G′ is finite. Later (in 1957), Wiegold found an explicit bound for the order of G′. We study groups in which the conjugacy classes containing commutators are finite with bounded size. We obtain the following results. Let G be a group and n a positive integer. If |xG|≤n for any commutator x∈G, then the second derived group G′′ is finite with n-bounded order. If |xG′|≤n for any commutator x∈G, then the order of γ3(G′) is finite and n-bounded. 1. Introduction Given a group G and an element x∈G, we write xG for the conjugacy class containing x. Of course, if the number of elements in xG is finite, we have ∣xG∣=[G:CG(x)]. A group is said to be a BFC-group if its conjugacy classes are finite and of bounded size. One of B. H. Neumann’s discoveries was that in a BFC-group, the derived group G′ is finite [3]. It follows that if ∣xG∣≤n for each x∈G, then the order of G′ is bounded by a number depending only on n. A first explicit bound for the order of G′ was found by J. Wiegold [7], and the best known was obtained in [1] (see also [4, 6]). In the present article, we deal with groups G such that ∣xG∣≤n whenever x is a commutator, that is, x=[x1,x2] for suitable x1,x2∈G. Here and throughout the article, we write [x1,x2] for x1−1x2−1x1x2. As usual, we denote by G′ the derived group of G and by G″ the derived group of G′ (the second derived group of G). Theorem 1.1. Let nbe a positive integer and Ga group in which ∣xG∣≤nfor any commutator x. Then ∣G″∣is finite and n-bounded. Further, we consider groups G in which ∣xG′∣≤n whenever x is a commutator. Theorem 1.2. Let nbe a positive integer and Ga group in which ∣xG′∣≤nfor any commutator x. Then ∣γ3(G′)∣is finite and n-bounded. Here γ3(G′) denotes the third term of the lower central series of G′. We do not know whether under hypothesis of Theorem 1.2, the second derived group G″ must necessarily be finite. Note that under hypothesis of Theorem 1.1, γ3(G) can be infinite. This can be shown using any example of an infinite torsion-free metabelian group whose commutator quotient is finite (see for instance [2]). We make no attempts to obtain good bounds for ∣G″∣ in Theorem 1.1 and ∣γ3(G′)∣ in Theorem 1.2. The proofs given here yield bounds n54n14 and n12n10, respectively. The bounds, however, do not look realistic at all. 2. Proofs Let G be a group generated by a set X such that X=X−1. Given an element g∈G, we write lX(g) for the minimal number l with the property that g can be written as a product of l elements of X. Clearly, lX(g)=0 if and only if g=1. We call lX(g) the length of g with respect to X. Lemma 2.1. Let Hbe a group generated by a set X=X−1and let Kbe a subgroup of finite index min H. Then each coset Kbcontains an element gsuch that lX(g)≤m−1. Proof If b∈K, the result is obvious. Therefore, we assume that b∈K. Choose g∈Kb in such a way that s=lX(g) is as small as possible and suppose that s≥m. Write g=x1⋯xs with xi∈X and set yj=x1⋯xj for j=1,…,s. Since s is the minimum of lengths of elements in Kb, it follows that none of the elements y1,…,ys lies in K. Thus, these s elements belong to the union of at most m−1 right cosets of K, and we conclude that Kyi=Kyj for some 1≤i<j≤s. It is now easy to see that the element h=yixj+1...xs belongs to Kb while lX(h)<lX(g). This is a contradiction with the choice of g.□ In the sequel, the above lemma will be used in the situation where H is the derived group of a group G and X is the set of commutators in G. Therefore, we will write l(g) to denote the smallest number such that the element g∈G′ can be written as a product of as many commutators. Recall that if H is a group and a∈H, the subgroup [H,a] is generated by all commutators of the form [h,a], where h∈H. It is well-known that [H,a] is always normal in H. Recall that in any group G, the following ‘standard commutator identities’ hold. [x,y]−1=[y,x]; [xy,z]=[x,z]y[y,z]; [x,yz]=[x,z][x,y]z. In what follows the above identities will be used without explicit references. We will now fix some notation and hypothesis. Hypothesis 2.2. Let Gbe a group and Ka subgroup containing H=G′. Let Xdenote the set of commutators in G, and suppose that CK(x)has finite index at most nin Kfor each x∈X. Let mbe the maximum of indices of CH(x)in H, where x∈X. Suppose further that a∈Xand CH(a)has index precisely min H. Choose b1,…,bm∈Hsuch that l(bi)≤m−1and aH={abi;i=1,…,m}. (The existence of such elements is guaranteed by Lemma2.1.) Set U=CK(⟨b1,…,bm⟩). Lemma 2.3. Assume Hypothesis2.2. Then for any x∈X, the subgroup [H,x]has finite m-bounded order. Proof Choose x∈X. Since CH(x) has index at most m in H, by Lemma 2.1 we can choose elements y1,…,ym such that l(yi)≤m−1 and [H,x] is generated by the commutators [yi,x]. For each i=1,…,m, write yi=yi1...yi(m−1), where yij∈X. The standard commutator identities show that [yi,x] can be written as a product of conjugates in H of the commutators [yij,x]. Let h1,…,hs be the conjugates in H of elements from the set {x,yij;1≤i,j≤m}. Since CH(h) has finite index at most m in H for each h∈X, it follows that s is m-bounded. Let T=⟨h1,…,hs⟩. It is clear that [H,x]≤T′ and so it is sufficient to show that T′ has finite m-bounded order. Observe that CH(hi) has finite index at most m in H for each i=1,…,s. It follows that the center Z(T) has index at most ms in T. Thus, Schur’s theorem [5, 10.1.4] tells us that T′ has finite m-bounded order, as required.□ Note that the subgroup U has finite n-bounded index in K. This follows from the facts that l(bi)≤m−1 and CK(x) has index at most n in K for each x∈X. The next lemma is somewhat analogous with Lemma 4.5 of Wiegold [7]. Lemma 2.4. Assume Hypothesis2.2. Suppose that u∈Uand ua∈X. Then [H,u]≤[H,a]. Proof Since u∈U, it follows that (ua)bi=uabi for each i=1,…,m. Therefore, the elements uabi form the conjugacy class (ua)H. For an arbitrary element g∈H, there exists h∈{b1,…,bm} such that (ua)g=uah and so ugag=uah. Therefore, [u,g]=aha−g∈[H,a]. The lemma follows.□ Proposition 2.5. Assume Hypothesis2.2and write a=[d,e]for suitable d,e∈G. There exists a subgroup U1≤Uwith the following properties. The index of U1 in K is n-bounded; [H,U1′]≤[H,a]d−1; [H,[U1,d]]≤[H,a]. Proof Set U1=U∩Ud−1∩Ud−1e−1. Since the index of U in K is n-bounded, we conclude that the index of U1 in K is n-bounded as well. Choose arbitrarily elements h1,h2∈U1. Write [h1d,eh2]=[h1,h2]d[d,h2][h1,e]dh2[d,e]h2 and so [h1d,eh2]h2−1=[h1,h2]dh2−1[d,h2]h2−1[h1,e]d[d,e]. Denote the product [h1,h2]dh2−1[d,h2]h2−1[h1,e]d by u. Thus, the right-hand side of the above equality is ua while, obviously, on the left-hand side we have a commutator. Let us check that u∈U. We see that [h1,h2]dh2−1∈U1dh2−1≤U because U1d≤U. By the same reason, [d,h2]h2−1∈U. Finally, [h1,e]d∈U1dU1ed≤U so indeed u∈U. By Lemma 2.4, [H,u]≤[H,a]. This holds for any choice of h1,h2∈U1. In particular, taking h1=1 we see that [H,[d,h2]h2−1]≤[H,a] while taking h2=1 we conclude that [H,[h1,e]d]≤[H,a]. It now follows that [H,[h1,h2]dh2−1]≤[H,a]. Since [H,a] is normal in H, we have [H,[h1,h2]]≤[H,a]d−1 and so [H,U1′]≤[H,a]d−1, which proves that U1 has Property 2. Examine again the inclusion [H,[d,h2]h2−1]≤[H,a]. Since [H,a] is normal in H, it follows that [H,[U1,d]]≤[H,a]. Therefore, U1 has Property 3 as well. The proof is now complete.□ We are ready to prove our main results. Proof of Theorem 1.1 Recall that G is a group in which ∣xG∣≤n for any commutator x. We need to show that ∣G″∣ is finite and n-bounded. We denote by X the set of commutators in G and set H=G′. Let m be the maximum of indices of CH(x) in H, where x∈X. Of course, m≤n. Choose a∈X such that CH(a) has index precisely m in H. Choose b1,…,bm∈H such that l(bi)≤m−1 and aH={abi;i=1,…,m}. Set U=CG(⟨b1,…,bm⟩). Note that the index of U in G is n-bounded. Applying Proposition 2.5 with K=G, we find a subgroup U1, of n-bounded index, such that [H,U1′]≤⟨[H,a]G⟩. Since the index of U1 in G is n-bounded, we can find n-boundedly many commutators c1,…,cs∈X such that H=⟨c1,…,cs,H∩U1⟩. Let T be the normal closure in G of the product of the subgroups [H,a] and [H,ci] for i=1,…,s. By Lemma 2.3, each of these subgroups has n-bounded order. Our hypothesis is that each of them has at most n conjugates. Thus, T is a product of n-boundedly many finite subgroups, normalizing each other and having n-bounded order. We conclude that T has finite n-bounded order. Therefore, it is sufficient to show that the second derived group of the quotient G/T has finite n-bounded order. So we pass to the quotient G/T. To avoid complicated notation, the images of G, H and X will be denoted by the same symbols. We observe that the derived group of HU1 is contained in Z(H). This follows from the facts that HU1 is generated by c1,…,cs and U1 and modulo T, we have c1,…,cs∈Z(H) and U1′≤Z(H). Let X denote the family of subgroups S≤G with the following properties: H≤S; S′≤Z(H); S has finite index in G. We already know that X is non-empty since it contains HU1. Choose J∈X of minimal possible index j in G. Since the index of U1 in G is n-bounded, the index j is n-bounded, too. We will now use induction on j. If j=1, then J=G and H≤Z(H). So G″=1 and we have nothing to prove. Thus, we assume that j≥2. Again, we take a commutator a0∈X such that CH(a0) has maximal possible index in H and write a0=[d,e] for suitable d,e∈G. If both d and e belong to J, we conclude (since J′≤Z(H)) that H is abelian and G″=1. Thus, assume that at least one of them, say d, is not in J. We will use Proposition 2.5 with K=G. It follows that there is a subgroup V of n-bounded index in G such that [H,[V,d]]≤[H,a0]. Replacing if necessary V by V∩J, without loss of generality we can assume that V≤J. Let L=J⟨d⟩. Note that L′=J′[J,d]. Let 1=g1,…,gt be a full system of representatives of the right cosets of V in J. Then [J,d] is generated by [V,d]g1,…,[V,d]gt and [g1,d],…,[gt,d]. This is straightforward from the fact that [vg,d]=[v,d]g[g,d] for any g,v∈G. Next, for each i=1,…,t set xi=[gi,d]. Let R be the normal closure in G of the product of the subgroups [H,a0]gi and [H,xi] for i=1,…,t. By Lemma 2.3, each of these subgroups has n-bounded order. Our hypothesis is that each of them has at most n conjugates. Thus, R is a product of n-boundedly many finite subgroups, normalizing each other and having n-bounded order. We conclude that R has finite n-bounded order. We see that [H,L′]≤R. Since d∉J, the index of L in G is strictly smaller than j. Therefore, by induction on j, the second derived group of G/R is finite with bounded order. Taking into account that also R is finite with bounded order, we deduce that G″ is finite with bounded order. The proof is now complete.□ Proof of Theorem 1.2 Recall that G is a group in which ∣xG′∣≤n for any commutator x. We need to prove that γ3(G′) is finite with n-bounded order. As before, we write X for the set of commutators in G and H for the derived group. Choose a commutator a∈X such that CH(a) has maximal possible index in H. We will use Proposition 2.5 with K=H. It follows that H contains a subgroup U1 of finite n-bounded index such that [H,U1′]≤[H,a]d−1 for some d∈G. Write b0=ad−1 and so [H,U1′]≤[H,b0]. Since the index of U1 in H is n-bounded, we can find n-boundedly many commutators b1,…,bs∈X such that H=⟨b1,…,bs,U1⟩. Let T be the product of the subgroups [H,bi] for i=0,1,…,s. By Lemma 2.3, each of these subgroups has n-bounded order. All of them are normal in H and so T is normal in H and has finite n-bounded order. The center of H/T contains the images of U1′ and b1,…,bs. It follows that the quotient of H/T over its center is abelian. Therefore, γ3(H)≤T, which completes the proof.□ Funding This research was supported by FAPDF and CNPq-Brazil. References 1 R. M. Guralnick and A. Maroti , Average dimension of fixed point spaces with applications , J. Algebra 226 ( 2011 ), 298 – 308 . 2 N. Gupta and S. Sidki , On torsion-free metabelian groups with commutator quotients of prime exponent , Int. J. Algebra Comput. 9 ( 1999 ), 493 – 520 . Google Scholar Crossref Search ADS 3 B. H. Neumann , Groups covered by permutable subsets , J. London Math. Soc. (3) 29 ( 1954 ), 236 – 248 . Google Scholar Crossref Search ADS 4 P. M. Neumann and M. R. Vaughan-Lee , An essay on BFC groups , Proc. Lond. Math. Soc. 35 ( 1977 ), 213 – 237 . Google Scholar Crossref Search ADS 5 D. J. S. Robinson , A course in the theory of groups , 2nd edn . Graduate Texts in Mathematics, 80. Springer-Verlag , New York , 1996 . 6 D. Segal and A. Shalev , On groups with bounded conjugacy classes , Quart. J. Math. Oxford 50 ( 1999 ), 505 – 516 . Google Scholar Crossref Search ADS 7 J. Wiegold , Groups with boundedly finite classes of conjugate elements , Proc. Roy. Soc. London Ser. A 238 ( 1957 ), 389 – 401 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
The Quarterly Journal of Mathematics – Oxford University Press
Published: Sep 1, 2018
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