A generalization of groups with many almost normal subgroups

A subgroup H of a group G is called almost normal in G if it has finitely many conjugates in G. A classic result of B. H. Neumann informs us that |G:Z(G)| is finite if and only if each H is almost normal in G. Starting from this result, we investigate the structure of a group in which each non-finit...

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Дата:	2010
Автор:	Russo, F.G.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2010
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154600
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Цитувати:	A generalization of groups with many almost normal subgroups / F.G. Russo // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 79–85. — Бібліогр.: 21 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1546002019-06-16T01:32:15Z A generalization of groups with many almost normal subgroups Russo, F.G. A subgroup H of a group G is called almost normal in G if it has finitely many conjugates in G. A classic result of B. H. Neumann informs us that \|G:Z(G)\| is finite if and only if each H is almost normal in G. Starting from this result, we investigate the structure of a group in which each non-finitely generated subgroup satisfies a property, which is weaker to be almost normal 2010 Article A generalization of groups with many almost normal subgroups / F.G. Russo // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 79–85. — Бібліогр.: 21 назв. — англ. 1726-3255 2010 Mathematics Subject Classification:20C07; 20D10; 20F24. http://dspace.nbuv.gov.ua/handle/123456789/154600 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	A subgroup H of a group G is called almost normal in G if it has finitely many conjugates in G. A classic result of B. H. Neumann informs us that \|G:Z(G)\| is finite if and only if each H is almost normal in G. Starting from this result, we investigate the structure of a group in which each non-finitely generated subgroup satisfies a property, which is weaker to be almost normal
format	Article
author	Russo, F.G.
spellingShingle	Russo, F.G. A generalization of groups with many almost normal subgroups Algebra and Discrete Mathematics
author_facet	Russo, F.G.
author_sort	Russo, F.G.
title	A generalization of groups with many almost normal subgroups
title_short	A generalization of groups with many almost normal subgroups
title_full	A generalization of groups with many almost normal subgroups
title_fullStr	A generalization of groups with many almost normal subgroups
title_full_unstemmed	A generalization of groups with many almost normal subgroups
title_sort	generalization of groups with many almost normal subgroups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/154600
citation_txt	A generalization of groups with many almost normal subgroups / F.G. Russo // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 79–85. — Бібліогр.: 21 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT russofg ageneralizationofgroupswithmanyalmostnormalsubgroups AT russofg generalizationofgroupswithmanyalmostnormalsubgroups
first_indexed	2025-07-14T06:39:00Z
last_indexed	2025-07-14T06:39:00Z
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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 Volume 9 (2010). Number 1. pp. 79 – 85 c© Journal “Algebra and Discrete Mathematics” A generalization of groups with many almost normal subgroups Francesco G. Russo Communicated by L. A. Kurdachenko Dedicated to Professor I.Ya. Subbotin on the occasion of his 60-th birthday Abstract. A subgroup H of a group G is called almost normal in G if it has finitely many conjugates in G. A classic result of B. H. Neumann informs us that \|G : Z(G)\| is finite if and only if each H is almost normal in G. Starting from this result, we investigate the structure of a group in which each non- finitely generated subgroup satisfies a property, which is weaker to be almost normal. 1. Anti-XC-Groups In this paper X denotes an arbitrary class of groups which is closed with respect to forming subgroups and quotients, F is the class of all finite groups, Fπ is the class of all finite π-groups (π set of primes), Č is the class of all Chernikov groups, PF is the class of all polycyclic-by-finite groups, S2F is the class of all (soluble minimax)-by-finite groups. Given a positive integer r, we recall that the operator L, defined by (1.1) LX = {G \| 〈g1, g2, . . . , gr〉 ∈ X, ∀g1, g2, . . . gr ∈ G}, from X to X is called local operator for X. See [12, §C, p.54]. We recall that the operator H, which associates to X the class of hyper-X-groups This paper is dedicated to the memory of my father and to the future of my brother. 2010 Mathematics Subject Classification: 20C07; 20D10; 20F24. Key words and phrases: Dietzmann classes; anti-XC-groups; groups with X- classes of conjugate subgroups; Chernikov groups. Jo u rn al A lg eb ra D is cr et e M at h .80 A generalization of groups with many ... is called extension operator. See [12, §E, p.60]. The notation follows [11, 12, 13, 16]. A subgroup H of a group G is called almost normal in G if H has finitely many conjugates in G, that is, if \|G : NG(H)\| is finite. Neumann’s Theorem [16, Chapter 4, Vol.I, p.127] shows that G has each H which is almost normal in G if and only if G/Z(G) ∈ F. We have NG(ClG(H)) = coreG(NG(H)) = ⋂ x∈G NG(H)x = ⋂ x∈G NG(H x), where ClG(H) is the set of conjugates of H in G. \|G : NG(H)\| = \|ClG(H)\| is finite if and only if G/coreG(NG(H)) ∈ F. In [8, 9] G has F-classes of conjugate subgroups, if G/coreG(NG(H)) ∈ F for each H in G. Thus Neumann’s Theorem can be reformulated, stating that G has G/coreG(NG(H)) ∈ F for each H in G if and only if G/Z(G) ∈ F. See [9, Introduction]. More generally, G has X-classes of conjugate subgroups, if G/coreG(NG(H)) ∈ X for each H in G. [9, Main Theorem] describes groups having Č-classes of conjugate subgroups. [8, Main Theorem] describes those having PF- classes of conjugate subgroups. Recall that ZX(G) = {x ∈ G \| G/CG(〈x〉 G) ∈ X} is a characteristic subgroup of G, called XC-center of G. See [12, Definition B.1, Proposi- tion B.2]. G is called XC-group if it coincides with its XC-center. FC- groups, ČC-groups, (PF)C-groups and (S2F)C-groups are well–known and described in [4, 7, 11, 12, 13, 15]. If G has F-classes of conjugate subgroups, then it is an FC-group. From [9, Lemma 2.3 ], if G has Č-classes of conjugate subgroups, then it is a ČC-group. From [8, Corollary 2.7], if G has PF-classes of conjugate subgroups, then it is a (PF)C-group. From [17, Lemma 2.4], if G has S2F-classes of conjugate subgroups, then it is an (S2F)C-group. The next lemma allows us to generalize these facts. Lemma 1.1. Assume that FX = X. If G has X-classes of conjugate subgroups, then ZX(G) = G. Proof. Let g ∈ G. G/H ∈ X, where H = coreG(NG(〈g〉)). Let H1 = CH(〈g〉) and H2 = coreG(H1) = CH(〈g〉G). It is enough to prove G/H2 ∈ X. Of course, H ≥ NH(〈g〉). Conversely, an element of NH(〈g〉) is an element of G, fixing 〈g〉x = 〈gx〉 by conjugation for every x ∈ G, again fixing 〈g〉 by conjugation. If x = 1, then we get the elements of H and so H ≤ NH(〈g〉). Then H/H1 = NH(〈g〉)/CH(〈g〉) is isomor- phic to a subgroup of the automorphism group of 〈g〉 and so it is finite. The same is true if we consider H1/H2 and NG(〈g〉)/CG(〈g〉). Therefore, G/H2 is an extension of the finite group H1/H2 by the finite group H/H1 by G/H ∈ X. From (FF)X = FX = X, G/H2 ∈ X. Jo u rn al A lg eb ra D is cr et e M at h .F. G. Russo 81 We recall that X is called Dietzmann class, if for every group G and x ∈ G, the following implication is true: (1.2) if x ∈ ZX(G) and 〈x〉 ∈ X, then 〈x〉G ∈ X, See [12, Definitions B.1 and B.6]. Dietzmann classes are studied in [11, 12, 13]. FC-groups form a Dietzmann class [12, Proposition D.3, b)]. In particular, this is true for periodic (PF)C-groups, which are obviously FC-groups. Note that F is a Dietzmann class [12, Proposition B.7, b)], but PF is not a Dietzmann class [12, Example B.8, c)]. Unfortunately, it is not known whether (PF)C-groups, ČC-groups or (S2F)C-groups form a Dietzmann class. See [4, 7, 11, 12, 13, 15]. But, they extend locally the class of FC-groups. Therefore, the next result is significant. Theorem 1.2 (see [12], Theorem E.3). If Fπ ⊆ X ⊆ LFπ, then (HX)C is a Dietzmann class. From Lemma 1.1, if X = F, then FC is a Dietzmann class. From Lemma 1.1 and Theorem 1.2, if Fπ ⊆ X ⊆ LFπ, then (HX)C is a Diet- zmann class. Therefore, it is meaningful to ask whether we may weaken the Neumann’s Theorem, looking at the following property for G: (1.3) if H is non-finitely generated, then G/coreG(NG(H)) ∈ X, where Fπ ⊆ X ⊆ LFπ. G is called anti-XC-group if it satisfies (1.3). Anti-FC-groups were de- scribed in [5]. Anti-ČC-groups and anti-(PF)C-groups were described in [18]. This line of research goes back to [14] and deals with the struc- ture of groups with given properties of a system of subgroups. See [1, 2, 3, 5, 6, 10, 18, 20, 21]. 2. Locally Finite Case We omit the elementary proofs of the next two results. Lemma 2.1. Subgroups and quotients of anti-XC-groups are anti-XC- groups. Lemma 2.2. If G is an anti-XC-group and ZX(G) = G, then G has X-classes of conjugate subgroups. Lemma 2.3. Assume that x is an element of the anti-XC-group G. If A = Dri∈IAi is a subgroup of G consisting of 〈x〉-invariant nontrivial direct factors Ai, i ∈ I, with infinite index set I, then x belongs to ZX(G). Jo u rn al A lg eb ra D is cr et e M at h .82 A generalization of groups with many ... Proof. This follows by [18, Lemma 3.3, Proof], considering X and ZX(G). Corollary 2.4. Assume that G is an anti-XC-group and A = Dri∈IAi is a subgroup of G consisting of infinitely many nontrivial direct factors. Then A is contained in ZX(G). Lemma 2.5. Assume that g is an element of the anti-XC-group G and A = Dri∈IAi is a subgroup of G, with I as in Lemma 2.3. If g ∈ NG(A) and gn ∈ CG(A) for some positive integer n, then g belongs to ZX(G). Proof. This follows by [18, Lemma 3.7, Proof], considering X and ZX(G). Corollary 2.6. If the anti-XC-group G has an abelian torsion subgroup that does not satisfy the minimal condition on its subgroups, then all elements of finite order belong to ZX(G). Proof. This follows by [18, Corollary 3.9, Proof], considering X and ZX(G). Theorem 2.7. If G is a locally finite anti-XC-group, then either G has X-classes of conjugate subgroups or G is a Chernikov group. Proof. This follows by [18, Theorem 3.12, Proof], considering X and ZX(G). Note that Theorem 2.7 improves [18, Theorems 3.11 and 3.12]. Lemma 2.8. Assume that X is residually closed. If G has X-classes of conjugate subgroups, then G ∈ N2X, where N2 is the class of nilpotent groups of class at most 2. Proof. Let N(G) = ⋂ H≤G NG(H) be the norm of G. N(G) ≤ Z2(G) from a result of Schenkman [19, Corollary 1.5.3]. Since G has X-classes of conjugate subgroups, G/N(G) is residually X and so G/N(G) ∈ X. This gives as claimed. Corollary 2.9. Assume that X is residually closed. If G is a locally finite anti-XC-group, then either G ∈ N2X or G is a Chernikov group. Proof. This follows by Theorem 2.7 and Lemma 2.8. Jo u rn al A lg eb ra D is cr et e M at h .F. G. Russo 83 3. Locally Nilpotent Case Recall that G has finite abelian section rank if it has no infinite elemen- tary abelian p-sections for every prime p (see [16, Chapter 10, vol.II]). Following [5, 16, 20], a soluble-by-finite group G is an S1-group if it has finite abelian section rank and the set of prime divisors of orders of elements of G is finite. Theorem 3.1. Assume that X is residually closed. Let G be an anti-XC- group having an ascending series whose factors are either locally nilpotent or locally finite. Then either G has X-classes of conjugate subgroups or is a soluble-by-finite S1-group or has a normal soluble S1-subgroup K such that G/K ∈ X. Proof. G has an ascending normal series whose factors are either locally nilpotent or locally finite by [16, Theorem 2.31]. Let K be the largest radical normal subgroup of G. From Lemma 2.1 and Corollary 2.9, the largest locally finite normal subgroup T/K of G/K is either a Chernikov group or in N2X. In the first case, if H/T is a locally nilpotent normal subgroup of G/T , then CH/K(T/K) is a locally nilpotent normal subgroup of G/K, so CH/K(T/K) is trivial and H/K is a Chernikov group. Then T = G and so G has a normal radical subgroup K such that T/K is a Chernikov group (in this situation G is said to be a radical-by-Chernikov group). In the second case, T/K = (N/K)(L/K), where N/K ∈ N2 is a normal subgroup of T/K such that (T/K)/(N/K) ∈ X. If N/K is nontrivial, then there exists a nontrivial element xK ∈ N/K such that 〈xK〉G = 〈x〉GK/K is a nilpotent normal subgroup of G/K contained in T/K. Since G/K has no nontrivial locally nilpotent normal subgroups, we get to a contradiction. Therefore N/K is trivial and T/K ∈ X. Then we may deduce as above that G has a normal radical subgroup K such that T/K ∈ X (in this situation G is said to be a radical-by-X group). Assume that G has X-classes of conjugate subgroups. Then every abelian subgroup of G has finite total rank by Corollary 2.4. A result of Charin [16, Theorem 6.36] implies that K is a soluble S1-group. We conclude that G has a normal soluble S1-subgroup K such that G/K is a Chernikov group. Therefore G is an extension of a soluble S1-group by an abelian group with min by a finite group. An abelian group with min is clearly an S1-group and the class of S1-groups is closed with respect to extensions of two of its members (see [16, Chapter 10]). Therefore G is a soluble-by-finite S1-group. The remaining case is that G has a normal soluble S1-subgroup K such that G/K ∈ X. Note that Theorem 3.1 improves [18, Theorems 4.1 and 4.2]. Jo u rn al A lg eb ra D is cr et e M at h .84 A generalization of groups with many ... Corollary 3.2. Assume that X is residually closed. Let G be an anti-XC- group having an ascending series whose factors are either locally nilpotent or locally finite. Then either G ∈ N2X or G is a soluble-by-finite S1-group or G has a normal soluble S1-subgroup K such that G/K ∈ X. Proof. This follows by Theorem 3.1 and Corollary 2.9. References [1] V. S. Charin, D. I. Zaitsev, Groups with finiteness conditions and other restric- tions for subgroups, Ukrainian Math. J., 40, 1988, pp.233–241. [2] S. N. Chernikov,Groups with given properties of a system of subgroups, Modern Algebra, Nauka, Moscow, 1980. [3] B. Hartley, A dual approach to Chernikov modules, Math. Proc. Cambridge Phil. Soc., 82, 1977, pp.215–239. [4] S. Franciosi, F. de Giovanni, M. J. Tomkinson, Groups with polycyclic-by-finite conjugacy classes, Boll. U.M.I., 4B, 1990, pp.35–55. [5] S. Franciosi, F. de Giovanni, L. A. Kurdachenko, On groups with many almost normal subgroups, Ann. Mat. Pura Appl., CLXIX, 1995, pp.35–65. [6] H. Heineken, L. A. Kurdachenko, Groups with Subnormality for All Subgroups that Are Not Finitely Generated, Ann. Mat. Pura Appl., CLXIX, 1995, pp.203– 232. [7] L. A. Kurdachenko, On groups with minimax conjugacy classes, In: Infinite groups and adjoining algebraic structures, Kiev (Ukraine), Naukova Dumka, 1993, pp.160–177. [8] L. A. Kurdachenko, J. Otál, P. Soules, Groups with polycyclic-by-finite conjugate classes of subgroups, Comm. Algebra, 32, 2004, pp.4769–4784. [9] L. A. Kurdachenko, J. Otál, Groups with Chernikov classes of conjugate sub- groups, J. Group Theory, 54, 2005, pp.93–108. [10] L. A. Kurdachenko, J. M. Munoz Escolano, J. Otál, Antifinitary linear groups, Forum Math., 20, 2008, pp.27—44. [11] R. Maier, J. C. Rogério XC-elements in groups and Dietzmann classes, Beiträge Algebra Geom., 40, 1999, pp.243–260. [12] R. Maier, Analogues of Dietzmann’s Lemma, In: Advances in Group Theory, Naples (Italy), Aracne Ed., 2002, pp.43–69. [13] R. Maier, The Dietzmann property of some classes of groups with locally finite conjugacy classes, J. Algebra, 277, 2004, pp.364–369 [14] G. A. Miller, H. C. Moreno Non-abelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc., 4, 1903, pp.398–404. [15] Ya. D. Polovickii, The groups with extremal classes of conjugate elements, Siberian Math. J., 5, 1964, pp.891–895. [16] D. J. Robinson. Finiteness conditions and generalized soluble groups, Vol. I and II, Springer, Berlin, 1972. [17] F.G. Russo, Groups with soluble minimax conjugate classes of subgroups, Mashhad Research J. Math. Sci., 1, 2007, pp.41–49. Jo u rn al A lg eb ra D is cr et e M at h .F. G. Russo 85 [18] F.G. Russo, Anti-CC-Groups and Anti-PC-Groups, Int. J. Math. Math. Sciences, Article ID 29423, 2007, 11 pages. [19] R. Schmidt, Subgroup lattices of groups, de Gruyter, Berlin, 1994. [20] D. I. Zaitsev, On locally soluble groups with finite rank, Doklady A. N. SSSR, 240, 1978, pp.257–259. [21] D. I. Zaitsev, On the properties of groups inherited by their normal subgroups, Ukrainian Math. J., 38, 1986, pp.707–713. Contact information F. G. Russo Department of Mathematics, University of Naples Federico II, via Cinthia I-80126, Naples, Italy E-Mail: francesco.russo@dma.unina.it URL: russodipmatunina.altervista.org Received by the editors: 25.02.2010 and in final form 25.02.2010.

A generalization of groups with many almost normal subgroups

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