Groups with many generalized FC-subgroup

Let FC⁰ be the class of all finite groups, and for each non-negative integer n define by induction the group class FCⁿ⁺¹ consisting of all groups G such that the factor group G/CG(xG) has the property FCⁿ for all elements x of G. Clearly, FC¹ is the class of FC-groups and every nilpotent group with...

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Datum:	2009
Hauptverfasser:	Russo, A., Vincenzi, G.
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spelling	irk-123456789-1546432019-06-16T01:31:06Z Groups with many generalized FC-subgroup Russo, A. Vincenzi, G. Let FC⁰ be the class of all finite groups, and for each non-negative integer n define by induction the group class FCⁿ⁺¹ consisting of all groups G such that the factor group G/CG(xG) has the property FCⁿ for all elements x of G. Clearly, FC¹ is the class of FC-groups and every nilpotent group with class at most m belongs to FCm. The class of FCⁿ-groups was introduced in [6]. In this article the structure of groups with finitely many normalizers of non-FCⁿ-subgroups (respectively, the structure of groups whose subgroups either are subnormal with bounded defect or have the property FCⁿ) is investigated. 2009 Article Groups with many generalized FC-subgroup / A. Russo, G. Vincenzi // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 158–166. — Бібліогр.: 17 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20F24. http://dspace.nbuv.gov.ua/handle/123456789/154643 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let FC⁰ be the class of all finite groups, and for each non-negative integer n define by induction the group class FCⁿ⁺¹ consisting of all groups G such that the factor group G/CG(xG) has the property FCⁿ for all elements x of G. Clearly, FC¹ is the class of FC-groups and every nilpotent group with class at most m belongs to FCm. The class of FCⁿ-groups was introduced in [6]. In this article the structure of groups with finitely many normalizers of non-FCⁿ-subgroups (respectively, the structure of groups whose subgroups either are subnormal with bounded defect or have the property FCⁿ) is investigated.
format	Article
author	Russo, A. Vincenzi, G.
spellingShingle	Russo, A. Vincenzi, G. Groups with many generalized FC-subgroup Algebra and Discrete Mathematics
author_facet	Russo, A. Vincenzi, G.
author_sort	Russo, A.
title	Groups with many generalized FC-subgroup
title_short	Groups with many generalized FC-subgroup
title_full	Groups with many generalized FC-subgroup
title_fullStr	Groups with many generalized FC-subgroup
title_full_unstemmed	Groups with many generalized FC-subgroup
title_sort	groups with many generalized fc-subgroup
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/154643
citation_txt	Groups with many generalized FC-subgroup / A. Russo, G. Vincenzi // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 158–166. — Бібліогр.: 17 назв. — англ.
series	Algebra and Discrete Mathematics
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2009). pp. 158 – 166 c⃝ Journal “Algebra and Discrete Mathematics” Groups with many generalized FC-subgroup Alessio Russo and Giovanni Vincenzi Communicated by I. Ya. Subbotin To Professor L. A. Kurdachenko on his sixties birthday Abstract. Let FC0 be the class of all finite groups, and for each non-negative integer m define by induction the group class FCm+1 consisting of all groups G such that the factor group G/CG(x G) has the property FCm for all elements x of G. Clearly, FC1 is the class of FC-groups and every nilpotent group with class at most m belongs to FCm. The class of FCm-groups was intro- duced in [6]. In this article the structure of groups with finitely many normalizers of non-FCm-subgroups (respectively, the struc- ture of groups whose subgroups either are subnormal with bounded defect or have the property FCm) is investigated. Introduction The structure of groups for which the set of non-normal subgroups (or more generally of non-subnormal subgroups) has prescribed properties has been investigated by several authors. The first step was of course the description of groups in which all subgroups are normal (Dedekind groups). It is well known that Dedekind groups either are abelian or can be decomposed as a direct product of Q8 (the quaternion group of order 8) and a periodic abelian group with no elements of order 4. G. M. Romalis and N. S. Sesekin ([14], [15], [16]) considered locally soluble groups whose non-normal subgroups are abelian (metahamiltonian groups), proving in particular that commutator subgroup of these groups is finite. Moreover, S. N. Chernikov [2] studied the groups whose non-normal subgroups are 2000 Mathematics Subject Classification: 20F24. Key words and phrases: Conjugacy class, FC-groups, normalizer subgroup, subnormal subgroup. Jo u rn al A lg eb ra D is cr et e M at h .A. Russo, G. Vincenzi 159 finite, while recently L. A. Kurdachenko, J. Otal and the authors [7] have described (generalized) soluble groups whose non-normal subgroups are FC-groups. They prove, among other results, that such groups are locally (central-by-finite). Finally, B. Bruno and R. E. Phillips [1] are concerned with (generalized) soluble groups whose non-normal subgroups are locally nilpotent, showing in particular that the groups of this class have finite commutator subgroup if they are not locally nilpotent. In a famous paper of 1955, B. H. Neumann [10] proved that each subgroup of a group G has finitely many conjugates if and only if the factor group G/Z(G) is finite, and the same conclusion holds if the restriction is imposed only to abelian subgroups (see [3]). Therefore central-by-finite groups are precisely those groups in which all the normalizers of (abelian) subgroups have finite index, and this result suggests that the behaviour of normalizers has a strong influence on the structure of the group. In this context, it is in- teresting to recall a result of Y. D. Polovickĭi [12] stating that a group G has finitely many normalizers of abelian subgroups if and only if it is central-by-finite. These considerations show that it is natural to investi- gate the structure of groups with finitely many normalizers of subgroups with a given property. For instance, F. De Mari and F. de Giovanni re- cently have studied the classes of groups with finitely many normalizers of non-abelian (respectively, of non-(locally nilpotent)) subgroups (see [4], [5]) improving the quoted result of Romalis and Sesekin (respectively, of Bruno and Phillips). The next step to combine the above ideas seems to be the considera- tion of classes of groups which are a good extension both of the nilpotency and of the property FC. To this purpose we will concerned with the class of FCm-groups introduced in [6]. Recall that FC0 is the class of all fi- nite groups, and for each non-negative integer m define by induction the group class FCm+1 consisting of all groups G such that for every element x ∈ G the factor group G/CG(x G) has the property FCm. Clearly, the FC1-groups are precisely the FC-groups while the class FCm contains all finite groups and all nilpotent groups with class at most m. The aim of this article is to investigate the structure of the classes of (generalized) soluble groups which either have finitely many normalizers of non-FCm- subgroups or have the non-subnormal subgroups with bounded defect satisfying the property FCm. Most of our notation is standard and can for instance be found in [9] and [13]. 1. Statementes and proofs We will prove many of our results within the universe of W -groups. Recall that a group G is said to be a W -group if every finitely generated non- Jo u rn al A lg eb ra D is cr et e M at h .160 Groups with many generalized FC-subgroups nilpotent subgroup of G has a finite, non-nilpotent, homomorphic image. It is well known that all hyper-(abelian or finite) groups and all linear groups have the property W (see [13], theorem 10.51 and [17]). Let k and m be non-negative integers. In what follows we denote by Xm and Xk,m respectively, the class of groups with finitely many normalizers of non-FCm-groups and the class of groups whose subgroups either are subnormal with defect at most k or satisfy the property FCm. Proposition 1. Let G be a W -group whose non-subnormal subgroups are FCm-groups for some non-negative integer m. Then every finitely generated subgroup of G is polycyclic-by-finite. Proof. Clearly we may suppose that G is a finitely generated non-nilpo- tent group. It follows that there exists a non-subnormal subgroup H of G with finite index, so that the core HG of H in G is a finitely generated FCm-group. Therefore the factor group HG/Zm(HG) is finite (see [6], proposition 3.6), and hence G contains a subgroup of finite index which is nilpotent with class at most m. In order to prove the corresponding result for the class Xm we need a lemma which has a formulation and proof similar to those obtained for other classes of groups with finitely many normalizer of subgroups of certain types (see for instance [4], [5], [8]). Lemma 1. Let G be an Xm-group. Then G contains a characteristic X2,m-subgroup of finite index. Proof. Let H be a subgroup of G which has not the property FCm. Clearly its normalizer NG(H) has finitely many conjugates in G, and hence the index ∣G : NG(NG(H))∣ is finite. It follows that the characteri- stic subgroup M(H) = ∩ �∈AutG NG(NG(H))� has finite index. Let L be the set of all non-FCm-subgroups of G. If H and K are elements of L such that NG(H) = NG(K), then M(H) = M(K). Thus also M = ∩ H∈L M(H) is a characteristic subgroup of finite index of G. Now let H be any element of L contained in M . Then M ≤ M(H) ≤ NG(NG(H)), and hence H is subnormal in M with defect at most 2. Proposition 2. Let G be a W -group satisfying the property Xm. Then every finitely generated subgroup of G is polycyclic-by-finite. Jo u rn al A lg eb ra D is cr et e M at h .A. Russo, G. Vincenzi 161 Proof. The statement immediately follows from Proposition 1 and Lem- ma 1. Our next results are turned to prove that the W -groups in the classes Xm and Xk,m have the torsion subgroup. To this end we need to inves- tigate the behaviour of the finitely generated torsion-free abelian normal subgroups of these groups. We begin with an easy result. Lemma 2. Let A be a torsion-free abelian group, and let g be a non- trivial automorphism of G. If g has finite order, then the semidirect product G = ⟨g⟩⋉A is not nilpotent. Proof. Assume, for a contradiction that G is nilpotent. Let giā be an element of Z(G) ∖A, where ā ∈ A and ∣g∣ does not divide i. As (giā)a = a(giā) for each a ∈ A, then gi ∈ Z(G), a contradiction because the centralizer C⟨g⟩(A) is trivial. Therefore Z(G) ≤ A, and MacLain theorem yields that G is torsion-free. Lemma 3. Let G be is any Xk,m-group (any Xm-group, respectively), and let A be a finitely generated torsion-free abelian normal subgroup of G. If g is an element of finite order of G, then [A, g] = {1}. Proof. Assume, for a contradiction that [A, g] ∕= {1}. Clearly, there exists a prime number p such that Yn = ⟨g⟩Apn is not abelian for all positive integers n. If Yn is an FCm-group, then m+1(Yn) is periodic (see [6], corollary 3.3) and hence Yn is nilpotent since A is torsion-free. Therefore, also the factor group Yn/C⟨g⟩(A pn) is nilpotent, a contradiction by the Lemma 2. Thus Yn is not an FCm-group for all n. First suppose that G is an Xk,m-group. Then Yn is subnormal in G with defect at most k for all positive integers n, whence so is the subgroup ⟨g⟩ = ∩ n∈N Yn. It follows by Fitting’s theorem that Yn is nilpotent, and we reach a con- tradiction using the argument above. Finally, let G be an Xm-group, and assume that G is a counterexample with a minimal number t of proper normalizers of subgroups which are not FCm-groups. Clearly, we may suppose that t ≥ 1. Then every subgroup Yn is normal in G, since oth- erwise its normalizer would be a counterexample with less than t proper normalizers of non-FCm-subgroups. It follows that also the subgroup ⟨g⟩ is normal in G. Thus [A, g] ≤ A ∩ ⟨g⟩ = {1}, and this last contradiction completes the proof of the lemma. Jo u rn al A lg eb ra D is cr et e M at h .162 Groups with many generalized FC-subgroups Theorem 1. Let G be a W -group satisfying the property Xk,m (the prop- erty Xm, respectively). Then the elements of finite order of G form a (fully-invariant) subgroup of G. Proof. Let x and y be elements of finite orders of G. Clearly, we may assume that G = ⟨x, y⟩, and by Proposition 1 (by Proposition 2, respec- tively) G is polycyclic-by-finite. Moreover, as the largest periodic normal subgroup T of G is finite, replacing G by the factor group G/T , it can also be assumed without loss of generality that G has no periodic non- trivial normal subgroups. Let S be a soluble normal subgroup of G such that G/S is finite. For a contradiction, let G be infinite, and choose the soluble subgroup S with minimal derived length. If A is the smallest non-trivial term of the derived series of S, then A is torsion-free, so that by Lemma 3 A is contained in the centre Z(G) of G. On the other hand, by the minimal choice of G, the factor group G/A is finite, and hence G′ is likewise finite by Schur’s theorem. It follows that also G is finite, and this contradiction proves the statement. Recall that if H is a subgroup of a group G, the series of normal closures of H in G is defined inductively by the positions HG,0 = G and HG,k+1 = HHG,k for each non-negative integer k. Thus H is subnormal in G with defect k if and only if HG,k = H and k is the smallest non- negative integer with this property. Corollary 1. Let G be a W -group satisfying the property Xk,m. Then the subgroup f(k+m+1)+1(G) is periodic, where f is the function of Rose- blade’s theorem. Proof. By Theorem 1 we can assume that G is torsion-free. It follows that every subgroup of G either is subnormal with defect at most k or is nilpotent with class at most m (see [6], corollary 3.4), and hence G is locally nilpotent (see [11], theorem A). Suppose that G is finitely gener- ated, and let X be a subgroup of G which is not subnormal with defect at most k. Then X < XG,k. Clearly, if M is a maximal subgroup of XG,k containing X, then M is nilpotent with class at most m, and hence X is subnormal in M with defect at most m. On the other hand, M ⊲ XG,k, so that X is subnormal in G with defect at most m + k + 1. Therefore the statement follows from the Roseblade’s theorem. Let H be a subgroup of a group G. Denote by IG(H) the isolator of H in G, i.e., the set of all x ∈ G such that xm ∈ H for some positive integer m. In general, IG(H) need not be a subgroup, as is shown by the isolator of the identity subgroup in D∞, the infinite dihedral group. On the other hand, if G is locally nilpotent, then the isolator of every subgroup of G is likewise a subgroup. Actually locally nilpotent groups have a rich isolator Jo u rn al A lg eb ra D is cr et e M at h .A. Russo, G. Vincenzi 163 theory. For details we refer to [9], Section 2.3. In particular, in what follows we will use the statement which ensures that if H is a subgroup of a torsion-free, locally nilpotent group G, then Zm(H) = Zm(IG(H))∩H for all non-negative integers m. Corollary 2. Let G be a W -group satisfying the property Xm. Then the subgroup m+1(G) is periodic. Proof. By Theorem 1 we can assume that G is torsion-free, and hence by corollary 3.4 of [6] it has finitely many normalizers of subgroups which are not nilpotent with class at most m. It follows that G is locally nilpotent (see [5], theorem A), and we may also suppose that G is finitely generated. Now we argue by induction on the number of proper norma- lizers of subgroups with class greater that m, and so we may reduce to the case that all non-normal subgroups of G have class at most m. Suppose that G does not contain a subgroup of finite index with class at most m. Let L be the set of all subgroups of finite index of G. Then each element K of L is normal in G and G/K is a Dedekind group. It follows that 3(G) ≤ ∩ K∈L K = {1}. Thus G is metahamiltonian and so even abelian, a contradiction. Now let H be a normal subgroup of G with class at most m such that the factor group G/H is finite. Clearly, the isolator IG(H) of H in G is equal to G. It follows by the quoted remark that H ≤ Zm(G), and hence G is nilpotent with class at most m as required. Corollary 3. Let G be a W -group satisfying the property Xm. If H is a finitely generated subgroup of G, then the factor group H/Zm(H) is finite. Proof. By Proposition 2 and Corollary 2 the subgroup m+1(H) is finite, whence so is H/Zm(H). Note that it is easy to find X1,1-groups with an infinite commutator subgroup which are not FC-groups. For, let p be a prime number, and let P be a group of type p∞. Consider the automorphism x of P × P defined by the rule x(a1, a2) = (a1a2, a2) for every (a1, a2) ∈ P×P . Then the semidirect product G = ⟨x⟩⋉ (P × P ) is a non-periodic group whose commutator subgroup is G′ = Z(G) ≃ P . Therefore, each non-normal subgroup of G has finite commutator subgroup. Moreover, G is not an FC-group since G/Z(G) is not periodic. Jo u rn al A lg eb ra D is cr et e M at h .164 Groups with many generalized FC-subgroups In [7], lemma 2.7, it has been proved that an X1,1-group has countable commutator subgroup if it is not an FC-group. Our last result extends this statement to the class Xm. For this purpose we need two lemmas concerning the FCm-groups. Lemma 4. Let G be an FCm-group. Then every non-trivial normal subgroup N of G contains a non-trivial FC-element of G. Proof. Clearly, we may suppose m ≥ 2 and [N,G] ∕= {1}. Denote by F the FC-centre of G. By theorem 3.2 of [6] the subgroup m(G) is contained in F so that [N,m−1G] ≤ N ∩F . It follows that there exists a positive integer n ≤ m− 1 such that {1} ∕= [N,nG] ≤ N ∩ F . Recall that a group class X is said to be countably recognizable if a group G belongs to X provided that all its countable subgroups are X- groups. In what follows we denote by FCm(G) the subgroup of a group G consisting of all elements x such that G/CG(⟨x G⟩) is an FCm−1-group. In particular, FC1(G) is the FC-centre of G. Lemma 5. Let G be a group, and let H be a countable subgroup of G. If every countable subgroup of G containing H is an FCm-group, then H ≤ FCm(G). In particular, the class FCm is countably recognizable for all non-negative integers m. Proof. Put N = FCm(G)∩HG, and let K/N be any countable subgroup of G/N containing HN/N . Then K = LN , where L is a countable sub- group of G, and H is contained in the countable FCm-subgroup ⟨H,L⟩. It follows that K/N ≤ ⟨H,L⟩N/N , and hence K/N is an FCm-group. Therefore the hypotheses are inherited by the group G/N and its count- able subgroup HN/N . Assume for a contradiction that the statement is false, so that H is not contained in N . Replacing G by G/N it can be assumed without loss of generality that the normal closure HG contains no non-trivial elements having finitely many conjugates in G. Write H = {ℎn ∣ n ∈ N0} where ℎ0 = 1, and put X0 = {1}. Now suppose that for some non- negative integer n a countable subgroup Xn of G has been defined con- taining the elements ℎ0, . . . , ℎn. Clearly there exists a countable sub- set Wn of G such that every non-trivial element of HG∩Xn has infinitely many conjugates under the action of Wn. Consider the countable sub- group Xn+1 = ⟨Xn, ℎn+1,Wn⟩, and put X = ∪ n∈N0 Xn. Jo u rn al A lg eb ra D is cr et e M at h .A. Russo, G. Vincenzi 165 Thus X is a countable subgroup of G containing H, and hence X is an FCm-group. It follows from Lemma 4 that there exists a non-trivial element u of HX ∩FC1(X). If m is a positive integer such that u ∈ Xm, then u has infinitely many conjugates under the action of Xm+1, and this last contradiction proves the lemma. Theorem 2. Let G be an Xm-group. If G is not an FCm-group, then the commutator subgroup G′ of G is countable. Proof. Let NG(X1), . . . , NG(Xt) be the proper normalizers of non-FCm- subgroups of G. First assume that the set NG(X1) ∪ ⋅ ⋅ ⋅ ∪ NG(Xt) is properly contained in G. Let x be an element of G ∖ (NG(X1) ∪ ⋅ ⋅ ⋅ ∪NG(Xt)). Then each subgroup of G containing x either is normal or an FCm- group. Since G does not satisfy the property FCm, by Lemma 5 G contains a countable subgroup H which is not an FCm-group. Therefore the subgroup ⟨H,x⟩ is normal in G and G/⟨H,x⟩ is a Dedekind group. Thus ∣G′⟨H,x⟩/⟨H,x⟩∣ ≤ 2, and hence G′ is countable. Now suppose that G = NG(X1) ∪ ⋅ ⋅ ⋅ ∪NG(Xt). It follows from a result of B. H. Neumann that it is possible to omit from the above union any subgroup of infinite index (see [13], lemma 4.17), and hence we can write G = NG(Xi1) ∪ ⋅ ⋅ ⋅ ∪NG(Xis), where the index ∣G : NG(Xij )∣ is finite for all j = 1, . . . , s. Clearly, e- very NG(Xij ) has less than t proper normalizers of non-FCm-subgroups, so that arguing by induction (on t), we may assume that NG(Xij ) ′ is countable. On the other hand, as the index ∣G : NG(NG(Xij ) ′)∣ is fi- nite, then (NG(Xij ) ′)G is countable, whence so is the subgroup N = ⟨NG(Xi1) ′, . . . , NG(Xis) ′⟩G. Moreover, the factor group G/N has a finite covering consisting of abelian subgroups, and hence it is central-by-finite (see [13], theorem 4.16). It follows from Schur’s theorem that G′N/N is finite whence G′ is countable also in this case. References [1] B. Bruno, R.E. Phillips, Groups with restricted non-normal subgroups, Math. Z., N. 176, 1981, pp.199–221. [2] S. N. Chernikov, The groups with prescribed properties of family of infinite sub- groups, Ukrain. Math. J., N. 19, 1967, pp.111–131. Jo u rn al A lg eb ra D is cr et e M at h .166 Groups with many generalized FC-subgroups [3] I. I. Eremin, Groups with finite classes of conjugate abelian subgroups, Mat. Sb. N. 47, 1959, pp.45–54. [4] F. De Mari, F. de Giovanni, Groups with finitely many normalizers of non-abelian subgroups, Ricerche Mat. N. 55, 2006, pp.311–317. [5] F. De Mari, F. de Giovanni, Groups with finitely many normalizers of non- nilpotent subgroups, Math. Proc. Roy. Irish Acad., N. 105A(2), 2007, pp.143–152. [6] F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J., N. 28, 2002, pp.241–254. [7] L.A. Kurdachenko, J. Otal, A. Russo, G. Vincenzi, Groups whose non-normal subgroups have finite conjugacy classes, Math. Proc. Roy. Irish Acad. N. 104 A(2), 2004, pp.177–189. [8] L.A. Kurdachenko, A. Russo, G. Vincenzi, Groups without proper abnormal sub- groups, J. Group Theory N. 9, 2006, pp.507–518. [9] J. C. Lennox, D.J.S. Robinson, The Theory of Infinite Soluble Groups, Clarendon, Oxford, 2004. [10] B. H. Neumann, Groups with finite classes of conjugate subgroups, Math. Z. N. 63, 1955, pp.76–96. [11] R.E. Phillips, J. S. Wilson, On certain minimal conditions for infinite groups, J. Algebra N. 51, 1978, pp.41–68. [12] Y. D. Polovickĭi, Groups with finite classes of conjugate infinite abelian subgroups, Soviet Math. (Iz. VUZ) N. 24, 1980, pp.52–59. [13] D.J.S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Springer, Berlin 1972. [14] G.M. Romalis, N.F. Sesekin, Metahamiltonian Groups, Ural Gos. Univ. Zap., N. 5, 1966, pp.101–106. [15] G.M. Romalis, N.F. Sesekin, Metahamiltonian Groups II, Ural Gos. Univ. Zap., N. 6, 1968, pp.52–58. [16] G.M. Romalis, N.F. Sesekin, Metahamiltonian Groups II, Ural Gos. Univ. Zap., N. 7, 1969-1970, pp.195–199. [17] B. A. F. Wehrfritz, Frattini subgroups in finitely genereted linear groups, J. Lon- don Math. Soc., N. 43, 1968, pp.619–622. Contact information A. Russo Dipartimento di Matematica, Seconda Uni- versità di Napoli, Via Vivaldi 43, I - 81100 Caserta (Italy) E-Mail: alessio.russo@unina2.it G. Vincenzi Dipartimento di Matematica e Informat- ica, Università di Salerno, Via Ponte Don Melillo, I - 84084 Fisciano, Salerno (Italy) E-Mail: gvincenzi@unisa.it Received by the editors: 11.07.2009 and in final form 11.07.2009.

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