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Groups with Many Abelian Subgroups ∗ M View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 347 (2011) 83–95 Contents lists available at SciVerse ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Groups with many abelian subgroups ∗ M. De Falco a, F. de Giovanni a, , C. Musella a,Y.P.Sysakb,1 a Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I - 80126 Napoli, Italy b Institute of Mathematics, Ukrainian National Academy of Sciences, vul. Tereshchenkivska 3, 01601 Kiev, Ukraine article info abstract Article history: It is known that a (generalized) soluble group whose proper Received 12 April 2011 subgroups are abelian is either abelian or finite, and finite minimal Availableonline1October2011 non-abelian groups are classified. Here we describe the structure Communicated by Gernot Stroth of groups in which every subgroup of infinite index is abelian. © 2011 Elsevier Inc. All rights reserved. MSC: 20F22 Keywords: Minimal non-abelian group Infinite index subgroup 1. Introduction The structure of groups in which every member of a given system X of subgroups is abelian has been studied for several choices of the collection X. The first step of such investigation is of course the case of the system consisting of all proper subgroups: it is well known that any locally graded group whose proper subgroups are abelian is either finite or abelian, and finite minimal non-abelian groups are completely described. Here a group G is called locally graded if every finitely generated non-trivial subgroup of G contains a proper subgroup of finite index; the class of locally graded groups con- tains all locally (soluble-by-finite) groups and all residually finite groups, and so it is a large class of generalized soluble groups. On the other hand, the consideration of Tarski groups (i.e. infinite simple groups whose proper non-trivial subgroups have prime order) shows that there exist infinite minimal non-abelian groups. A further progress in this research area was due to G.M. Romalis and N.F. Sesekin, * Corresponding author. E-mail addresses: [email protected] (M. De Falco), [email protected] (F. de Giovanni), [email protected] (C. Musella), [email protected] (Y.P. Sysak). 1 This work was done while the last author was visiting the University of Napoli Federico II. He wishes to thank the G.N.S.A.G.A. of I.N.d.A.M. for financial support. 0021-8693/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2011.09.010 84 M. De Falco et al. / Journal of Algebra 347 (2011) 83–95 who considered groups in which all non-normal subgroups are abelian, and proved that (generalized) soluble groups with such property have finite commutator subgroups (see [6–8]). Recently, groups in which every non-abelian subgroup satisfies a given generalized normality condition have also been studied (see [1]). ∞ The aim of this paper is to describe the class A consisting of all groups in which every subgroup ∞ of infinite index is abelian. Of course, the class A is closed with respect to subgroups and homomor- phic images and contains all minimal non-abelian groups. Moreover, it is also clear that non-abelian ∞ groups in the class A are finitely generated and that every periodic subgroup of a non-periodic ∞ ∞ A -group is abelian. Note also that any A -group satisfies the maximal condition on non-abelian subgroups; information on the structure of these latter groups has been obtained by L.A. Kurdachenko and D.I. Zaicev [3]. ∞ Our main result on A - groups is the following. ∞ Theorem. Let G be an infinite non-abelian A -group, and let T be its largest periodic normal subgroup. Then G satisfies one of the following conditions: (a) G is finitely generated, G/Z(G) is a just-infinite group with no abelian subgroups of finite index and any two maximal abelian subgroups of G/Z(G) have trivial intersection. (b) G is soluble with derived length at most 3, T is finite abelian and one of the following statements holds: (b1) G =x T , where x is an element of infinite order such that [T , x]={1}. (b2) G =x (T ×y×z),wherex, y and z are elements of infinite order such that [T , x]={1}, [x, z]=1 and [x, y]=tzn for some t ∈ T and some positive integer n. (b3) G = (x(T ×y)g, where x and y are elements of infinite order such that [T , x]=[T , g]={1}, − − g3 ∈ T,1 = [x, y]∈T,xg = yandyg = zy 1x 1 for some z ∈ T. (b4) G = (x (T ×z×y)g, where [T , x]=[T , g]={1}, [z, x]=[z, g]=1, [x, y]=tzn for − − some element t of T and some positive integer n, g3 ∈ T z,xg = yandyg = uzm y 1x 1 for some element u of T and some non-negative integer m. (b5) G is isomorphic to a subgroup of a direct product T × K , where T is a finite abelian group and K =g A is a semidirect product of a torsion-free abelian normal subgroup A and a finite cyclic subgroup g whose non-trivial elements act rationally irreducibly on A. (b6) G is isomorphic to a subgroup of a direct product T × K , where T is a finite abelian group and K =g A is a semidirect product of a torsion-free abelian normal subgroup A and an infinite cyclic m n subgroup g such that Cg(A) =g for some integer m > 1 and g acts rationally irreducibly on A for each proper divisor n of m. Most of our notation is standard and can be found in [4]. 2. Preliminary results The first lemma is an obvious property of groups containing a cyclic subgroup of finite index; as ∞ a consequence, cyclic-by-finite A -groups are easily described. Lemma 2.1. Let G be a cyclic-by-finite group. Then G contains a finite normal subgroup N such that the factor group G/N either is cyclic or infinite dihedral. In particular, if G is not finite-by-cyclic, then it is generated by its finite subgroups. Corollary 2.2. Let G be a cyclic-by-finite infinite group and let T be its largest periodic normal subgroup. Then ∞ GbelongstotheclassA if and only if either T is abelian and G/T is infinite cyclic or T = Z(G) and G/Tis infinite dihedral. In the following, we shall denote by Zα(G) the α-th term of the upper central series of the group G, and by Z¯ (G) its hypercentre. Our next two results describes the behaviour of the upper ∞ central series of A -groups. M. De Falco et al. / Journal of Algebra 347 (2011) 83–95 85 ∞ Lemma 2.3. Let G be an A -group and let N be a normal subgroup of G. Then either N is contained in Z(G) or G/N is finite-by-cyclic. Proof. Suppose that N is not contained in Z(G). Of course, it can be assumed that G/N is infinite, so that in particular N is abelian. Consider an element x of G such that x, N is not abelian; then the index |G :x, N| is finite and hence G/N is cyclic-by-finite. If G/N is generated by its finite subgroups, we have that G is generated by its abelian subgroups containing N, and so N Z(G), a contradiction. Thus G/N is not generated by its finite subgroups, and hence G/N is finite-by-cyclic by Lemma 2.1. 2 ∞ ¯ Corollary 2.4. Let G be an A -group. Then either Z(G) = Z2(G) or the factor group G/Z(G) is finite. Proof. Suppose that Z(G) is properly contained in Z¯ (G).ThenG/Z¯ (G) is finite-by-cyclic by Lemma 2.3, and so G/Z¯ (G) is finite since it has trivial centre. 2 ∞ It is now possible to describe torsion-free nilpotent A -groups. The proof of our next lemma makes use of the well-known result by A.I. Mal’cev stating that if G is a torsion-free nilpotent group, then the factor group G/Z(G) is likewise torsion-free (see [4, Part 1, p. 53]). ∞ Lemma 2.5. Let G be a torsion-free nilpotent non-abelian A -group. Then G =x y×z , where [x, z]=1 and [x, y]=zn for some positive integer n. Proof. Since G is not abelian, there exist elements x and y such that xy = yx and x, y has class 2. Then x, y has finite index in G, so that G itself has class 2 and [x, y] belongs to Z(G).AsG/Z(G) is torsion-free, it follows that the index |Z(G) :[x, y]| is finite and hence Z(G) is cyclic; let z be the generator of Z(G) such that [x, y]=zn for some integer n > 0. Clearly, the factor group G/Z(G) is free abelian of rank 2, and so the elements x and y can be chosen in such a way that G =x, yZ(G). Therefore G has the structure described in the statement. 2 Recall that a group is said to be minimax if it has a series of finite length whose factors either satisfy the minimal or the maximal condition on subgroups. It was proved by P.H. Kropholler [2] that a finitely generated soluble group either is minimax or has a section which is isomorphic to the standard restricted wreath product of a group of prime order by an infinite cyclic group.
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