On some developments in investigation of groups with prescribed properties of generalized normal subgroups

The survey is dedicated to investigation of groups with prescribed properties of generalized normal subgroups. The roots of such investigations lie in the works by R. Dedekind, R. Baer, O.Yu.Schmidt, and S.N. Chernikov. The main goal of this survey is to reflect some important developments in this a...

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Дата:	2010
Автори:	Kirichenko, V.V., Kurdachenko, L.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2010
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	On some developments in investigation of groups with prescribed properties of generalized normal subgroups / V.V. Kirichenko, L.A. Kurdachenko // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 41–71. — Бібліогр.: 118 назв. — англ.

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spelling	irk-123456789-1544912019-06-16T01:32:04Z On some developments in investigation of groups with prescribed properties of generalized normal subgroups Kirichenko, V.V. Kurdachenko, L.A. The survey is dedicated to investigation of groups with prescribed properties of generalized normal subgroups. The roots of such investigations lie in the works by R. Dedekind, R. Baer, O.Yu.Schmidt, and S.N. Chernikov. The main goal of this survey is to reflect some important developments in this area. 2010 Article On some developments in investigation of groups with prescribed properties of generalized normal subgroups / V.V. Kirichenko, L.A. Kurdachenko // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 41–71. — Бібліогр.: 118 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20E15,20E34,20F19,20F22. http://dspace.nbuv.gov.ua/handle/123456789/154491 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	The survey is dedicated to investigation of groups with prescribed properties of generalized normal subgroups. The roots of such investigations lie in the works by R. Dedekind, R. Baer, O.Yu.Schmidt, and S.N. Chernikov. The main goal of this survey is to reflect some important developments in this area.
format	Article
author	Kirichenko, V.V. Kurdachenko, L.A.
spellingShingle	Kirichenko, V.V. Kurdachenko, L.A. On some developments in investigation of groups with prescribed properties of generalized normal subgroups Algebra and Discrete Mathematics
author_facet	Kirichenko, V.V. Kurdachenko, L.A.
author_sort	Kirichenko, V.V.
title	On some developments in investigation of groups with prescribed properties of generalized normal subgroups
title_short	On some developments in investigation of groups with prescribed properties of generalized normal subgroups
title_full	On some developments in investigation of groups with prescribed properties of generalized normal subgroups
title_fullStr	On some developments in investigation of groups with prescribed properties of generalized normal subgroups
title_full_unstemmed	On some developments in investigation of groups with prescribed properties of generalized normal subgroups
title_sort	on some developments in investigation of groups with prescribed properties of generalized normal subgroups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/154491
citation_txt	On some developments in investigation of groups with prescribed properties of generalized normal subgroups / V.V. Kirichenko, L.A. Kurdachenko // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 1. — С. 41–71. — Бібліогр.: 118 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 SURVEY ARTICLE Volume 9 (2010). Number 1. pp. 41 – 71 c© Journal “Algebra and Discrete Mathematics” On some developments in investigation of groups with prescribed properties of generalized normal subgroups V. V. Kirichenko and L. A. Kurdachenko Dedicated to Professor I.Ya. Subbotin on the occasion of his 60-th birthday Abstract. The survey is dedicated to investigation of groups with prescribed properties of generalized normal subgroups. The roots of such investigations lie in the works by R. Dedekind, R. Baer, O.Yu.Schmidt, and S.N. Chernikov. The main goal of this survey is to reflect some important developments in this area. 1. Introduction One of the main fruitful trends in group theory is defined by investigations of groups with prescribed properties of subgroups. With researches in this area, many important notions such as finiteness conditions, locally nilpo- tency, locally solubility, ranks, and many others have been introduced. The influence of these ideas is difficult to overestimate. Imposing of some natural restrictions on specifically chosen families of subgroups, we de- fine concrete classes of groups satisfying having these properties. Among many others, the following restrictive properties have been employed by numerous authors: normality, generalized normality, to be abelian, nilpo- tency, complementability, transitivity, supersolubility, density, the min- imal and maximal conditions, restrictions on important characteristics of groups ( in particular, on distinct ranks ), other finiteness conditions. Topological and linear groups with prescribed restrictions on families of subgroups have been also investigating. 2000 Mathematics Subject Classification: 20E15,20E34,20F19,20F22. Key words and phrases: normal subgroups, arrangement of subgroups, fan subgroups, pronormal subgroups, abnormal subgroups, quasicental subgroups. Jo u rn al A lg eb ra D is cr et e M at h .42 On some developments in investigation of groups In this survey, we will consider some important developments in in- vestigation of groups with prescribed properties for subgroups that are related in some way to generalized normality. Of course, the choice of the topics has been determined not only by logic of the subject but also by the authors’ tests. The roots of these investigations lie in the famous classical paper due to R. Dedekind [31], in which he completely described the finite non-abelian groups whose all subgroups are normal (the Hamiltonian groups). Recall that abelian and Hamiltonian groups together form the (named in honor of Dedekind) class of Dedekind groups. Later, R. Baer obtained a description of all infinite and finite Hamiltonian groups [8]. As it has been shown, such groups are direct products of a quaternion group, an elementary abelian 2 – group, and an abelian periodic group with elements of only odd order. Reducing the system of subgroups or weakening the restricted conditions we obtain some classes of groups close to or quite far from the class of Hamiltonian groups. Thus O. Yu. Schmidt has described all groups having only one class of non-normal subgroups [110], and all groups with two classes of non-normal subgroups [111]. As a proof of nowadays actuality of these Schmidt’s results, we can mention that these researches were continued in [114] and quite recently have been simply repeated in [15] and [76], and directly generalized in [80]. Some important results due to B. Huppert, N. Ito, J. Szep, Z. Janko, Z., J. Buckly, and many others (see respectively [38, 44, 45, 46, 47, 5]) are closely related to this. In infinite groups, this approach was transformed in the following way: the groups whose family of non-normal subgroups is in some sense very small became the objects of consideration. Thus, S.N.Chernikov studied groups whose all infinite subgroups are normal, groups whose all non-abelian subgroups are normal [25], [26], and groups whose infi- nite abelain subgroups are normal [27]. F.N. Liman described periodic groups whose all non-cyclic subgroups normal and groups with all abelain non-cyclic subgroups are normal [72], [73], [74]. G.M. Romalis and N.F. Sesekin investigated some groups in which every non-normal subgroup satisfies a given condition. In particular, in the papers [88], [89], [90], they considered groups in which non-normal subgroups are abelian (so called the metahamiltonian groups). The detailed description of metahamil- tonian groups was obtained in a series of papers due to N.F. Kuzenny and N.N. Semko (see the book [57]). L.A. Kurdachenko, J. Otal, Russo and Vinchenci recently investigated the class of groups whose non-FC- subgroups are normal. Clearly, this class contains the classes of meta- hamiltonian and minimal non-FC-groups (i.e. the non-FC-groups with all proper subgroups being FC-groups [53]). Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 43 Of course, this list could be significantly extended since it is not pos- sible to reflect all developments in this are in one survey. So we will focus on some interesting specific aspects of these studies that have not been reflected in surveys yet. 2. The Baer norm of a group and the quasicentralizer con- ditions Starting with the concept of Dedekind groups we naturally come to the following generalization of the concept of the center of a group, namely to the Baer norm of a group, which is defined in the following equivalent ways: It is the intersection of normalizers of all its subgroups, it is the intersection of normalizers of all cyclic subgroups. We denote the Baer norm of a group G by BN(G). Clearly, BN(G) is a Dedekind group, therefore BN(BN(G)) = BN(G). R.Baer introduced and during 1934-1956 investigated this concept in the cycle of his works [9, 10, 11, 12, 13, 14]. In particular, he proved that even though the norm of a group includes the group’s center, but in the case of periodic groups, it could be Hamiltonian. He also observed that for a group with the identity center, the norm is also identity [14]. L. Wos and E. Shenkman [117], [91] established that the second hypercenter of a group includes the norm of this group. Let A be a subgroup of a group G and let L(A) be a family of all sub- groups of A. Put QCG(A) = ⋂ H∈L(A) NG(H). This subgroup is called a norm of the subgroup A in the group G (see, for example, [116]). Among others, this concept plays an important role in the investigation of nor- mally factorized groups (see, for example, [118]). The norm of a subgroup A is also called the invariator or the quasicentralizer of A. We will use the last term in this survey. We say that a subgroup A is quasicentral in a group G if every sub- group of A is normal in G. The concept of quasicental subgroup became important in relation with investigations of groups with transitivity of normality. These studies take a special central place in research of groups with prescribed properties of normal subgroups. A group G is said to be a T−group if every subnormal subgroup of G is normal. A group G is said to be a T̄ -group, if every subgroup of G is a T−group. It should be noted that T−groups are the relatively old objects of investigation (see, for example, [16, 36, 1, 83, 40, 39, 109]). The structure of finite soluble T−groups has been described by W. Gaschütz [36]. In particular, Jo u rn al A lg eb ra D is cr et e M at h .44 On some developments in investigation of groups he found that every finite soluble T−group is a T̄−group. Observe that a finite T̄−group is metabelian. Infinite soluble T−groups and infinite T̄ -groups have been considered by D.J.S. Robinson [83]. Locally soluble T̄ -group G has the following structure. 2.1. D.J.S. Robinson [83]. Let G be a locally soluble T̄ – group. (i) If G is not periodic, then G is abelian. (ii) If G is periodic and L is the locally nilpotent residual of G, then G satisfies the following conditions: a) G/L is a Dedekind group; b) π(L) ∩ π(G/L) = ∅; c) 2 /∈ π(L); d) and every subgroup of L is G−invariant. In particular, if L 6= 〈1〉, then L = [L,G]. It easily follows that every subgroup of a derived subgroup of a soluble T -group is normal in the group. In another words, the derived subgroup of a soluble T -group is quasicentral in a group. A group G is called a KI−group, if every subgroup of [G,G] is G−invariant (I.Ya. Subbotin [92]). It is obvious that KI-groups are direct and quite wide generaliza- tions of soluble T -groups. For example, any group with cyclic derived subgroup is a KI -group. It is interesting to admit that KI-groups have abelian derived subgroups. As in the study of finite T -groups, it was not difficult to describe finite non-nilpotent KI-groups. 2.2. (I.Ya. Subbotin [92]). Let G be a finite non–nilpotent group. Then G is a KI−group if and only if G satisfies the following conditions: (i) G = L⋋H where L is abelian and H is nilpotent; (ii) [H,H] is normal in G; (iii) π(L) ∩ π([H,H]) = ∅; (iv) every subgroup of L is G – invariant. However, since all groups with a cyclic derived subgroups and all groups of nilpotency class ≤ 2 are KI–groups, there is impossible to obtain a detailed description of nilpotent KI–groups. These finite groups have been considered in [93]. Infinite finitely generated KI−groups have been studied in [94]. The most general results on KI-groups have been obtained in [98] and [99]. We collect together the main results of these papers. 2.3. Let G be an arbitrary group with a Chernikov derived subgroup. The group G is a KI-group if and only if the following conditions hold: (i) G = A ⋋ B where A is an abelian Chernikov subgroup, B is a hypercentral group; (ii) every subgroup of A is G−invariant; Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 45 (iii) [B,B] is normal in G; (iv) π(A) ∩ π(B́) = ∅. If A is a hypercentral rezidula of G,then all complements to A in G are conjugate. 2.4. Let G be an arbitrary group with a non-periodic derived subgroup whose periodic part is Chernikov. The group G is a KI-group if and only if G is either a nilpotent group of class 2, or G satisfied the following conditions : (i) G = A⋋B where A is an abelian Chernikov subgroup, B is a KI- group; (ii) every subgroup of A is G-invariant; (iii) 2 /∈ π(A); (iv) [B,B] is normal in G; (v) [B,B] = C×D where C is an abelian torsion-free group, D is an abelian Chernikov 2-group or an identity; (vi) B/C is hypercentral; (vii) \|G : CG(Ǵ)\| ≤ 2; (viii) (Ǵ ∩ ζ(G))2 = 〈1〉 ; (vii) any two distinct involutions from [G,G] are reciprocal. 2.5. Let G be a hypercentral KI-group. (1) If [G,G] is non-priodic, then G is nilpotent of class 2. (2) Suppose that [G,G] is periodic and denote by Dp the Sylow p- subgroup of [G,G]. Then (i) if the orders of elements of Dp∩ζ(G) are not bounded, then Dp ≤ ζ(G); (ii) if exp(Dp) = pn = exp(Dp ∩ ζ(G)), then Dp ≤ ζ(G); (iii) if exp(Dp) = pn, exp(Dp ∩ ζ(G)) = pm,n > m, and p 6= 2, then G/CG(Dp) is a cyclic group of order pn−m; (iv) if exp(D2) = 2n, exp(Dp ∩ ζ(G)) = 2m, and n > m, then G/CG(D2) is a subgroup of a direct product of a group of order 2 and a cyclic group of order 2n−m; (v) if the orders of elements of Dp are not bounded and exp(Dp ∩ ζ(G)) = pm, then G/CG(Dp) is a subgroup of the multiplicative group of p-adic numbers of the form ατ = 1 + ∑ i≥m xτi p i where 0 ≤ xτi < p, i ≥ 2, and there exists such ατ that xτm 6= 0. In particular, if p 6= 2 or p = 2 and m > 1, then G/CG(Dp) is torsion free; if p = 2 and m = 1, then Tor(G/CG(D2)) is a subgroup of order 2. In [99], the following results concerning KI−p-groups have been also obtained. 2.6. Let G be a p−group where p is a prime. Jo u rn al A lg eb ra D is cr et e M at h .46 On some developments in investigation of groups (1) Suppose that the orders of elements of [G,G] are not bounded. Then G is a KI− group if and only if G either is a nilpotent group of class at most 2, or G/CG([G,G]) = 〈xCG([G,G])〉 is a group of order 2. In the last case, p = 2 and cx = c−1 for each c ∈ CG([G,G]). (2) Suppose that exp([G,G]) = pn, exp([G,G] ∩ ζ(G)) = pm, n > m. Then G is a KI− group if and only if the following conditions hold: (i) G/CG([G,G]) is a finite abelian group; (ii) if p 6= 2, then G/CG([G,G]) is a cyclic group of order pn−m; (iii) if p = 2, then G/CG([G,G]) is a subgroup of a direct product of a group of order 2 and a cyclic group of order 2n−m; (iv) for every coset CG([G,G])x there is a positive integer tx such that yx = ysx where sx = 1 + txp m. Moreover, there is a coset CG([G,G])z such that (tz, p) = 1. In this connection, it is interesting to note that V. P. Shunkov consid- ered groups that decomposed into uniform product of Sylow p−subgroups [115]. Following Shunkov, we say that a group G is decomposed into uniform product of subgroups Hλ, λ ∈ Λ, if 〈x〉〈y〉 = 〈y〉〈x〉 for every el- ements x ∈ Hλ, y ∈ Hµ, for each pair of indexes λ, µ ∈ Λ. V.P. Shunkov described periodic groups that decomposed into uniform product of its Sylow p− subgroups. 2.7. ( V.P. Shunkov [115]). Let G be a group. Then G is decomposed into uniform product of its Sylow p−subgroups for all prime p if and only if G = A ⋋ B where A is a normal abelian subgroup, A is quasicentral in G, B = ∏ p∈π(B) Bp, Bp is a Sylow p− subgroup of B, p ∈ π(B), π(A) ∩ π(B) = ∅. In this setting in [99] it was established that a periodic KI-group G with a Chernikov derived subgroup is decomposed into uniform product of its Sylow p-subgroups if and only if its hypercentral residual is a Hall subgroup of G. The following concept of a quisecentral product of groups are natural generalizations of KI-groups [100], [102]. We call a group G a quasicen- tral product of a subgroup A by a subgroup B if G = AB and B ≤ QCG(A) [104]. This concept is a generalization of a similar concept of a quisicen- tral extension of a group [100]. As examples of quisicentral products of groups we can mention periodic groups decomposed into uniform product of its Sylow p−subgroups [115], soluble T -groups [83], and KI-groups. In [104] the following result has been obtained. This result is especially in- teresting in view of the question of complementing of the hypercentral residual in KI− and T−groups. Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 47 2.8. (I.Ya. Subbotin [104]) Let G be a quicentral product of a periodic hypercentral subgroup A by a periodic hypercentral subgroup B. Let A =∏ p∈π(A) Ap, where Ap is a Sylow p−subgroup of A, p ∈ π(B), and let Γ = {p ∈ π(A) \| Ap ∩ ζ(G) = 〈1〉}. Then the following assertions hold: (i) C = ∏ p∈Γ Ap is a hypercentral residual of G; (ii) C is an abelian subgroup and 2 /∈ π(C); (iii) C has a complement in G. This result has been generalized in a following way 2.9. (I.Ya. Subbotin [102]) Let G = AB be a periodic group where A,B be hypercentral subgroups satisfying the following conditions: A =∏ λ∈Λ Aλ, and B ≤ QCG(Aλ) for each λ ∈ Λ. Let ∆ = {λ ∈ Λ \| Aλ ∩ ζ(G) = 〈1〉}. Then the following assertions hold: (i) C = ∏ λ∈∆ Aλ is a hypercentral residual of G; (ii) C is an abelian subgroup and 2 /∈ π(C); (iii) C has a complement in G. This result has direct relation to the main results on the groups with all complemented subgroups [42, 20, 21, 22]. In [37] a well known sophisti- cated construction has been developed. This construction, in particular, allows to construct examples of periodic groups that are non-splitting extensions of its abelian Hall derived subgroup by an uncountable ele- mentary abelian 2-group. In the paper [66], this constraction has been extended on on the non-periodic case. In this paper, the authors, in particularly, constructed examples of non-periodic soluble T -groups and KI-groups with non-complemented periodic hypercentral residual. The qusicentralizer condition is directly connected to these topic. This is a strong form of the well known normalizer condition which plays one of the main roles in generalized nilpotency. We will say that a group G satisfies the quisicentralizer condition on the family of subgroups A if QCG(A)\A 6= ∅ for each subgroup A ∈ A [95]. It is almost obvious, that the groups with quasicentralizer condition on all subgroups are Dedekind groups. In the cycle of work [95, 96, 97, 101, 103, 105]. I. Ya. Sub- botin completely described all groups with the quasicentralizer condition on normal subgroups. Observe, that this groups form a subclass of T - groups. We collect together the main results of these papers. 2.10. Let G be a periodic soluble group with quaisicentralizer con- dition on normal subgroups. Then G is a group of one of the following types of groups. Jo u rn al A lg eb ra D is cr et e M at h .48 On some developments in investigation of groups I. G = C × ( ∏ λ∈Λ Aλ ⋋ 〈bλ〉) (i) C is an abelian subgroup; (ii) a set Λ is countable or finite; (iii) Aλ is a non-identity abelian subgroup for each λ ∈ Λ; (iv) bλ is a pλ−element where pλ is a prime, λ ∈ Λ; (v) 2, pµ /∈ π(Aλ) for all µ, λ ∈ Λ, µ 6= λ; (vi) pµ 6= pλ whenever µ 6= λ; (vii) π(Aλ) ∩ π(Aµ) = ∅ whenever µ 6= λ; (viii) every subgroup of Aλ is 〈bλ〉−invariant; (ix) bpλλ ∈ ζ(G) for each λ ∈ Λ. II. G = G1 ×G2 where (i) π(G1) ∩ π(G2) = ∅; (ii) G2 is a group of type I; (iii) G1 = (D ⋋ 〈t〉) × H, where D is an abelian subgroup without involution, \|t\| = 2, 4 and H is an elementary abelian 2 – group (H can be identity); (iv) dt = d−1 for all d ∈ D. III. G = G1 ×G2 where (i) π(G1) ∩ π(G2) = ∅; (ii) G2 is a group of type I; (iii) G1 = ((D ⋋ 〈t〉) × 〈s, z〉 × H, where D is an abelian subgroup without involution, \|t\| = 2, 〈z, s〉 is a quaternion group, and H is an elementary abelian 2−group (H can be identity); (iv) dt = d−1 for all d ∈ D. IV. G = G1 ×G2 where (i) π(G1) ∩ π(G2) = ∅; (ii) G2 is a group of type I; (iii) G1 = ((D × D0) ⋋ 〈t〉) × H, where D is an abelian subgroup without involution, D0 is a divisible abelian 2−subgroup, \|t\| = 2, 〈z, s〉 is a quaternion group, and H is an elementary abelian 2−group (H can be identity); (iv) dt = d−1 for all d ∈ D ×D0. V. G = G1 ×G2 where (i) π(G1) ∩ π(G2) = ∅; (ii) G2 is a group of type I; (iii) G1 = ((D×〈s〉) 〈z〉)×H, where D is an abelian subgroup without involution, 〈z, s〉 is a quaternion group, and H is an elementary abelian 2−group (H can be identity); (iv) dt = d−1 for all d ∈ D × 〈s〉 . Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 49 VI. G = G1 ×G2 where (i) π(G1) ∩ π(G2) = ∅; (ii) G2 is a group of type I; (iii) G1 = ((D × D0 × 〈s〉) 〈z〉) × H, where D is an abelian sub- group without involution, D0 is a divisible abelian 2−subgroup, 〈z, s〉 is a quaternion group, and H is an elementary abelian 2−group (H can be identity); (iv) dt = d−1 for all d ∈ D ×D0 × 〈s〉 . VII. G = G1 ×G2 where (i) π(G1) ∩ π(G2) = ∅; (ii) G2 is a group of type I; (iii) G1 = ((D×S) 〈z〉)×H, where D is an abelian subgroup without involution, S is a quisicyclic 2−group, 〈S, z〉 is a locally quaternion group, and H is an elementary abelian 2−group (H can be identity); (iv) dt = d−1 for all d ∈ D × S. VIII. G = G1 ×G2 where (i) π(G1) ∩ π(G2) = ∅; (ii) G2 is a group of type I; (iii) G1 = ((D × D0 × S) 〈z〉) × H, where D is an abelian subgroup without involution, D0 is a divisible abelian 2−subgroup, S is a quisicyclic 2−group, 〈S, z〉 is a locally quaternion group, and H is an elementary abelian 2−group (H can be identity); (iv) dt = d−1 for all d ∈ D ×D0 × S. Conversely, every of groups of the types I – VIII satisfies the quasi- centralizer condition on normal subgroups. 2.11. Let G be a non-periodic soluble group with the quasicentralizer condition on normal subgroups. Then G is a group of one of the following types of groups. I. G is an abelian group; II. G = C ×B where B is an elementary abelian 2−group (B can be identity), and C = D ⋋ 〈t〉, where D is a non-periodic abelian subgroup such that D = D2, \|t\| = 2, 4, and dt = d−1 for all d ∈ D. III. G = C × B where B is an elementary abelian 2−group (B can be identity), and C = D⋋ < 〈s, z〉 , where D is a non-periodic abelian subgroup such that D = D2, 〈s, z〉 is a quaternion group, 〈s〉 ∈ CG(D) and dz = d−1 for all d ∈ D. IV. G = C × B where B is an elementary abelian 2−group (B can be identity), and C = D⋋ < 〈S, z〉, where D is a non-periodic abelian subgroup such that D = D2, S is a quasicyclic 2−group, 〈S, z〉 is a locally quaternion group, S ∈ CG(D) and dz = d−1 for all d ∈ D. Jo u rn al A lg eb ra D is cr et e M at h .50 On some developments in investigation of groups Conversely, every of groups of the types I – IV satisfies the quasicen- tralizer condition on normal subgroups. 2.12. Let G be a periodic non–soluble group with the quasicentralizer condition on normal subgroups. Then G = Zrλ∈ΛGλ is a direct product with joint center of groups Gλ, and the following conditions hold. (i) there exists an index µ ∈ Λ such that [Gµ, Gµ] is abelian; (ii) for every λ ∈ Λ\ {µ} the factor-group Gλ/ζ(G) is a simple non- abelian group; (iii) if Dµ is the Sylow 2́-subgroup of Gµ, then π(Dµ)∩π(〈Gλ \| λ ∈ Λ\{µ}〉) = ∅; (iv) if Dµ 6= Gµ, then the subgroup 〈Gλ \| λ ∈ Λ\{µ}〉 has no elements of order 4. Conversely, the group having the above properties satisfies the quasi- centralizer condition on normal subgroups. 2.13. Let G be a non-periodic and non-soluble group with the quasi- centralizer condition on normal subgroups. Then G is a central extension of an abelian group by a group decomposed in a direct product of simple non – abelian groups. Conversely, every such group satisfies the quasicentralizer condition on normal subgroups. In [107], [108] some types of groups with the quasicentralizer condi- tions on non-abelian normal subgroups have been described. 3. Generalized normality and arrangement of subgroups Let G be a group and G0 its subgroup. A subgroup H is called interme- diate to G0 if G0 ≤ H ≤ G [2]. Z.I. Borevich and his students studied the lattices of all subgroups intermediate to a fixed subgroup G0. They gen- eralized the theorem on homomorphisms on some non-normal subgroups (see [18, 17, 7, 34, 2]). The following definition belongs to Z.I. Borevich [2]. Let G be a group and G0 a subgroup of G. We say that a family {Gλ \| λ ∈ Λ} of intermediate to G0 subgroups is a fan for G0 if for each intermediate subgroup H there exists an unique index λ(H) ∈ Λ such that Gλ(H) ≤ H ≤ NG(Gλ(H)). The factor-groups NG(Gλ(H))/Gλ(H) are called sections of this fan. Each normal subgroup G0 of G is a fan subgroup in G. In this case, the fan consists of this subgroup G0 only and has the unique section G/G0. An opposite example of fan subgroups is provided by a subgroup G0 whose every intermediate subgroup H is self-normalizing in G, i. e. Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 51 NG(H) = H. In this setting, the fan is the set of all intermediate to G0 subgroups, and all sections are trivial. We obtain a natural generalization of a Dedekind group if we suppose that in a group G there exists a fixed subgroup M(G) such that every non- normal subgroup D ≤ G has a fan {D,M(G)}. In the case M(G) = G, the following theorem has been obtained in [71] (note that in this setting every non-normal subgroup D is normal in every proper subgroup H ≥ D of G). 3.1. Let G be a locally graded group and suppose that each its subgroup D has a fan {D,G}. Then G is a group of one of the following types: (i) G is a Dedekind group; (ii) G is a finite non-abelian group whose proper subgroups are abelian; (iii) G = Q ⋋ C, where Q is a quaternion group, C is a cyclic p−subgroup, where p is an odd prime; (iv) G is a generalized quaternion group of order 16. It is important to note that A.Yu. Olshanskii [79, Theorems 28.1, 31.8] has constructed examples of infinite groups with two generators whose proper subgroups are abelian (even cyclic). These groups are not locally graded. A subgroup H of a group G is abnormal in G if g ∈ 〈H,Hg〉 for each element g of G. Abnormal subgroups have been introduced in the paper [43] due to P. Hall, while the term "abnormal subgroup" itself belongs to R. Carter [23]. Abnormal subgroups are antipodes to normal subgroups. If G0 is abnormal subgroup and H an intermediate subgroup to G0, then H is self-normalizing. In this setting, the set of all intermediate to G0 subgroups is a fan for G0. Hence every abnormal subgroup is a fan subgroup. A subgroup H of a group G is said to be pronormal in G if for ev- ery g ∈ G the subgroups H and Hg conjugate in the subgroup 〈H,Hg〉. These subgroups have been introduced by P. Hall [42]. Such important subgroups of finite (soluble) groups as Sylow subgroups, Hall subgroups, system normalizers, and Carter subgroups are pronormal. Based on these concepts J. Rose considered balanced chains of subgroups in a group and contranormal subgroups [86]. Later, Z.I. Borevich and his students [2] in- troduced some generalizations of the mentioned subgroups, namely poly- normal, paranormal, weakly pronormal, and weakly abnormal subgroups. All these definitions (except of the Carter subgroups) have no limitation of finiteness. Let D be a fan subgroup in G. If the family of all intermediate to D subgroups satisfies the minimal condition, then there exists a unique fan Jo u rn al A lg eb ra D is cr et e M at h .52 On some developments in investigation of groups for D [2]. In particular, the uniqueness of the fan holds for finite groups. Simple examples show that this statement is not true for infinite groups. However some subgroups such as pronormal and abnormal subgroups always have a unique fan [2]. These subgroups play an important role in arrangement of subgroups [18, 17, 7, 34, 2]. T.A. Peng has considered finite groups whose all subgroups are pronor- mal. He proved the following result. 3.2. (T.A. Peng [81]) Let G be a finite soluble group. Then every subgroup of G is pronormal is and only if G is a T−group. However, in the infinite case, as the following theorem shows, the situation is much more sophisticated. 3.3. (N.F. Kuzennyi and I.Ya. Subbotin [67] ). Let G be a locally soluble group or a periodic locally graded group. Then the following con- ditions are equivalent. 1. Every cyclic subgroup of G is pronormal in G. 2. G is a soluble T -group. Infinite groups whose subgroups are pronormal have been considered in [65]. The authors completely described such infinite locally soluble non-periodic and infinite locally graded periodic groups. The main result of this paper is the following interesting theorem. 3.4 (N.F. Kuzennyi and I.Ya. Subbotin [65]) Let G be a group whose all subgroups are pronormal, and L be a locally nilpotent residual of G. (i) If G is periodic and locally graded, then G is a soluble T̄– group, in which L complements every Sylow π(G/L)– subgroup. (ii) If G is non-periodic and locally soluble, then G is abelian. Conversely, if G has a such structure, then every subgroup of G is pronormal in G. In the paper [87], the assertion (ii) has been extended to non-periodic locally graded groups proving that in this case such groups still to be abelian. N.F. Kuzennyi and I. Ya. Subbotin also completely described locally graded periodic groups in which all primary subgroups are pronormal [69], and infinite locally soluble groups in which all infinite subgroups are pronormal [68]. They proved that in the infinite case, the class of groups whose subgroups are pronormal is a proper subclass of T̄ -groups. More- over, it is also a proper subclass of the class of groups whose primary subgroups are pronormal. However, it is important to admit that the pronormality condition for all subgroups can be weakened to the pronor- mality for only abelian subgroups [70]. Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 53 Let G be a group and D be its subgroup. An intermediate subgroup F, D ≤ F ≤ G, is called a complete intermediate subgroup if the normal closure DF of D in F coincides with F. A subgroup D is called a polynormal subgroup in a group G if for any x ∈ G the subgroup D<x> = 〈Dx \| x ∈ 〈x〉〉 is a complete intermediate subgroup [2]. From the fan point of view, these concepts could be characterized in the following way [2]. 3.5. (M.S. Ba, Z. I. Borevich Z. I. [2]) (i) D is polynormal in group G if and only if it is a fan subgroup and all complete intermediate subgroups form its fan. (ii) D is abnormal in group G if and only if a) D is a fan subgroup and its fan basis consists of all intermediate subgroups, and b) any two intermediate conjugate subgroups coincide. (iii) D is pronormal in G if and only if a) D is a fan subgroup and its fan basis consists of D and all subgroups of group G, which strictly contain the normalizer NG(D); and b) any such two conjugate subgroups coincide. The subgroups mentioned above and their generalizations are very useful in finite group theory. In infinite groups, these subgroups gain some properties they cannot posses in the finite case. For example, it is well-known that every finite p–group has no proper abnormal subgroups. Nevertheless, A.Yu. Olshanskii has constructed a series of examples of infinite finitely generated p–groups saturated with abnormal subgroups. Specifically, for a sufficiently large prime p there exists an infinite p-group G whose all proper subgroups have prime order p [79, Theorem 28.1]. In finite soluble groups, abnormality is tightly bounded to self-normalizing. For example, D. Taunt has shown that a subgroup H of a finite solu- ble group G is abnormal if and only if every intermediate subgroup for H coincides with its normalizer in G; that is, such a subgroup is self- normalizing (see, for example, [84, 9.2.11]). The following theorem extends this result to the radical groups [62]. 3.6. (L.A. Kurdachenko, I.Ya. Subbotin [62]) Let G be a radical group and let H be a subgroup of G. Then H is abnormal in G if and only if every intermediate subgroup for H is self-normalizing. As corollaries we obtain Jo u rn al A lg eb ra D is cr et e M at h .54 On some developments in investigation of groups 3.7 (F. de Giovanni, G. Vincenzi [35]) Let G be a hyperabelian group and let H be a subgroup of G. Then H is abnormal in G if and only if every intermediate subgroup for H is self-normalizing. 3.8. (M.S. Ba, Z. I. Borevich Z. I. [2]) Let G be a soluble group and let H be a subgroup of G. Then H is abnormal in G if and only if every intermediate subgroup for H is self-normalizing. Recall that a subgroup H of a group G is said to have the Frattini property, if given two intermediate subgroups K and L for H such that K E L, we have L ≤ NG(H)K (in this case, it is also said that H is weakly pronormal in G). It is not hard to see that every pronormal subgroup has Frattini property. 3.9. (T. A. Peng [82]) Let G be a finite soluble group and D be a subgroup of G. Then D is pronormal in G if and only if D has a Frattini property. This Peng’s characterization of pronormal subgroups could be ex- tended in the following way. Let X be a class of groups. Recall that a group G is said to be a hyper–X–group if G has an ascending series of normal subgroups whose factors are X–groups. Recall that a group G is an N−group if H 6= NG(H) for each subgroup H of G. 3.10. ( L.A. Kurdachenko, J. Otal, I.Ya. Subbotin [49]) Let G be a hyper–N–group. Then a subgroup H of G is pronormal in G if and only if H is weakly pronormal in G. As corollaries we obtain 3.11. (F. de Giovanni, G. Vincenzi [35]) Let G be a hyperabelian group and let H be a subgroup of G. Then H is pronormal in G if and only if H is weakly pronormal in G [35]. 3.12. Let G be a soluble group and let H be a subgroup of G. Then H is pronormal in G if and only if H is weakly pronormal in G. Carter subgroups are important case of abnormal subgroups. These subgroups have been introduced by R. Carter [23] as the self-normalizing nilpotent subgroups of a finite group. Some attempts of extending the definition of a Carter subgroup to infinite groups were made by S.E. Stonehewer [112, 113], A.D. Gardiner, B. Hartley and M.J. Tomkinson [33], and M.R. Dixon [30]. In [62], this concept have been extended to the class of nilpotent-by-hypercentral (not necessary periodic) groups. Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 55 We may define a Carter subgroup of a finite metanilpotent group as a minimal abnormal subgroup. The first logical step here is to consider those groups whose locally nilpotent residual is nilpotent. Let X be a class of groups. A group G is said to be an artinian-by-X- group if G has a normal subgroup H such that G/H ∈ X and H satisfies Min−G. 3.13. ( L.A. Kurdachenko, J. Otal, I.Ya. Subbotin [49]) Let G be an artinian–by–hypercentral group and suppose that its locally nilpotent residual K is nilpotent. (i) G has a minimal abnormal subgroup L. Moreover, L is maximal hypercentral subgroup, and it includes the upper hypercenter of G. In particular, G = KL. (ii) Two minimal abnormal subgroups of G are conjugate. Thus, given an artinian–by–hypercentral group G with a nilpotent hypercentral residual, a subgroup L is called a Carter subgroup of a group G if L is a hypercentral abnormal subgroup of G or equivalently, if H is a minimal abnormal subgroup of G. A Carter subgroup of a finite soluble group can be also character- ized as a covering subgroup for the formation of nilpotent groups. In the paper [62], this characterization was extended to the class of artinian– by–hypercentral groups with a nilpotent locally nilpotent residual. Following J.S. Rose [86], a subgroup H of a group G is called contra- normal, if HG = G. Abnormal subgroups are contranormal. If H is a pronormal subgroup and H ≤ L, then NL(H) is abnormal in L. A subgroup H of a group G is called nearly pronormal if NL(H) is contranormal in L for every subgroup L including H. In the paper [55], the groups whose subgroups are nearly pronormal have been considered. 3.14. ( L.A. Kurdachenko, A. Russo, G. Vincenzi [55]) Let G be a locally radical group. (i) If every cyclic subgroup of G is nearly pronormal, then G is a T̄– group. (ii) If every subgroup of G is nearly pronormal, then every subgroup of G is pronormal in G. If G is a finite group, then for each subgroup H there is a chain of subgroups H = H0 ≤ H1 ≤ . . . . ≤ Hn−1 ≤ Hn = G Jo u rn al A lg eb ra D is cr et e M at h .56 On some developments in investigation of groups such that Hj is maximal in Hj+1, 0 ≤ j ≤ n − 1. Generalizing this, J. Rose has arrived at the balanced chain connecting a subgroup H to a group G, that is, a chain of subgroups H = H0 ≤ H1 ≤ . . . . ≤ Hn−1 ≤ Hn = G such that for each j, 0 ≤ j ≤ n − 1, either Hj is normal in Hj+1, or Hj is abnormal in Hj+1; the number n is the length of this chain. He refers appropriately to two consecutive subgroups Hj ≤ Hj+1 as forming a normal link or an abnormal link of this chain [85]. In a finite group, every subgroup can be connected to the group by some balanced chain. It is natural to consider the case when all of these balanced chains are short, i.e. their lengths are bounded by a small number. If these lengths are ≤ 1, then every subgroup is either normal or abnormal in a group. Such finite groups were studied in [32]. Infinite groups of this kind and some of their generalizations were described in [106] and [28]. Moreover, in the last paper have been considered the groups, whose subgroups are either abnormal or subnormal. More general situation was considered in a paper of L.A. Kurdachenko and H. Smith [58] . They considered the groups, whose subgroups are either self – normalizing or subnormal. Observe that in the groups in which the normalizer of any subgroup is abnormal and in the groups in which every subgroup is abnormal in its normal closure, the mentioned lengths are ≤ 2. It is logical to choose these groups as the subject for investigation. It is interesting to observe that if G is a soluble T -group, then every subgroup of G is abnormal in its normal closure. As we mentioned above, for any pronormal subgroup H of a group G, the normalizer NG(H) is an abnormal subgroup of G. So the subgroups having abnormal normal- izers make a generalization of pronormal subgroups. There are examples showing that this generalization is non-trivial. The article [56] initiated the study of groups whose subgroups are connected to a group by balanced chains of length at most 2. As we recently mentioned, such groups are naturally related to the T -groups. 3.15. (L.A. Kurdachenko, A. Russo, I.Ya. Subbotin, G. Vincenzi [56]) Let G be a radical group. Then G is a T -group if and only if every cyclic subgroup of G is abnormal in its normal closure. 3.16. (L.A. Kurdachenko, A. Russo, I.Ya. Subbotin, G. Vincenzi [56]) Let G be a periodic soluble group. Then G is aT–group if and only if its locally nilpotent residual L is abelian and the normalizer of each cyclic subgroup of G is abnormal in G. The following result from [56] is a new interesting and useful charac- terization of groups with all pronormal subgroups. Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 57 3.17. (L.A. Kurdachenko, A. Russo, I.Ya. Subbotin, G. Vincenzi [56]) Let G be a periodic soluble group. Then every subgroup of G is pronormal if and only if its locally nilpotent residual L is abelian and the normalizer of every subgroup of G is abnormal in G. For the non-periodic case, there exist non-periodic non-abelian groups in which normalizers of all subgroups are abnormal [56]. On the other hand, the non-periodic locally soluble groups in which all subgroups are pronormal are abelian [65]. So, in the non-periodic case we cannot count on a characterization, similar to above. However, we have the following result. 3.18 (L.A. Kurdachenko, A. Russo, I.Ya. Subbotin, G. Vincenzi [56]) Let G be a non-periodic group with the abelian locally nilpotent residual L. If the normalizer of every cyclic subgroup is abnormal and for each prime p ∈ Π(L) the Sylow p-subgroup of L is bounded, then G is abelian. Following [59], we will call normal and abnormal subgroups U -normal (from ”union” and “U -turn”). Finite groups with only U -normal sub- groups have been considered in [32]. Locally soluble (in the periodic case locally graded) infinite groups with U -subgroups have been studied in [106]. In [59], the groups with all U -normal subgroups and the groups with transitivity of U -normality were completely described. Next natural question regarding the structure of groups whose U - normal subgroups form a lattice. These groups are denoted as #U -groups [64]. It is easy to see that the groups with no abnormal subgroups are #U -groups. In particular, all locally-nilpotent groups have this property [67]. Observe that a union of any two U -normal subgroups is U -normal. However, the similar assertion is obviously false for intersections. It is easy to see that in a soluble group an abnormal subgroup R is exactly the subgroup that is contranormal in all subgroups containing R [28]. The condition "every contranormal subgroup is abnormal" (the CA-property) is an amplification of the transitivity of abnormality (the TA-property). Some simple examples show that the class of TA-groups is wider then the class of CA-groups and does not coincide with the class of #U -groups. A description of soluble CA-groups having #U -property were ob- tained in [64]. Jo u rn al A lg eb ra D is cr et e M at h .58 On some developments in investigation of groups 4. Generalized normality and criteria of generalized nilpo- tency The following well-known characterizations of finite nilpotent groups are tightly bound to abnormal and pronormal subgroups. A finite group G is nilpotent if and only if G has no proper abnormal subgroups. A finite group G is nilpotent if and only if its every pronormal sub- group is normal. Note that since the normalizer of a pronormal subgroup is abnormal, the absence of abnormal subgroups is equivalent to the normality of all pronormal subgroups. One part of these criteria is still true for infinite groups. 4.1. (N.F. Kuzenny, I.Ya. Subbotin [68]) Let G be a locally nilpotent group. Then G has no proper abnormal subgroups and every pronormal subgroup of G is normal. However, we do not know whether or not the converse to this result holds. In the paper [48], the following generalization of a well-known nilpo- tency criterion was obtained. Let G be a group, A a normal subgroup of G. We say that A satisfies the condition Max−G (respectively Min−G) if A satisfies the maximal (respectively the minimal) condition for G–invariant subgroups. A group G is said to be a generalized minimax group, if it has a finite series of normal subgroups every factor of which is abelian and either satisfies Max−G or Min−G. Every soluble minimax group is obviously generalized minimax. How- ever, the class of generalized minimax groups is significantly wider than the class of soluble minimax groups. In the paper [48], the first generalization of the mentioned nilpotency criterion was obtained. 4.2. (L.A. Kurdachenko, J. Otal, I.Ya. Subbotin [48]) Let G be a soluble generalized minimax group. If every pronormal subgroup of G is normal (or, what is equivalent, G has no proper abnormal subgroups), then G is hypercentral. Let G be a group. Then the set FC(G) = {x ∈ G \| xG is finite} Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 59 is a characteristic subgroup of G which is called the FC–center of G. Note that a group G is an FC–group if and only if G = FC(G). Starting from the FC–center, we construct the upper FC–central series of a group G 〈1〉 = C0 ≤ C1 ≤ · · · ≤ Cα ≤ Cα+1 ≤ · · ·Cγ where C1 = FC(G), Cα+1/Cα = FC(G/Cα) for all α < γ, and FC(G/Cγ) = 〈1〉. The term Cα is called the α–FC–hypercenter of G, while the last term Cγ of this series is called the upper FC–hypercenter of G. If Cγ = G, then the group G is called FC–hypercentral, and, if γ is finite, then G is called FC–nilpotent. The following criteria of hypercentrality have been obtained in [54]. 4.3. (L.A. Kurdachenko, A. Russo, G. Vincenzi [54]) Let G be a group whose pronormal subgroups are normal. Then every FC−hypercenter of G having finite number is hypercentral. Let G be an FC–nilpotent group. If all pronormal subgroups in G are normal, then G is hypercentral. Let G be a group whose pronormal subgroups are normal. Suppose that H be an FC- hypercenter of G having finite number. If C is a normal subgroup of G such that C ≥ H and C/H is hypercentral, then C is hypercentral. For periodic groups, the above results were obtained in [59]. Observe that abnormal subgroups are an important particular case of contranormal subgroups: abnormal subgroups are exactly the subgroups that are contranormal in each subgroup containing them. On the other hand, abnormal subgroups are a particular case of pronormal subgroups. Pronormal subgroups are connected to contranormal subgroups in the following way. If H is a pronormal subgroup of a group G and H ≤ K, then its normalizer NK(H) in K is an abnormal and hence contranormal subgroup of K. Starting from the normal closure of H, we can construct the normal closure series of H in G HG = H0 ≥ H1 ≥ ...Hα ≥ Hα+1 ≥ ...Hγ by the following rule: Hα+1 = HHα for every α < γ, Hλ = ⋂ µ<γ Hµ for a limit ordinal λ. The term Hα of this series is called the α-th normal closure of H in G and will be denoted by HG,α. The last term Hγ of this Jo u rn al A lg eb ra D is cr et e M at h .60 On some developments in investigation of groups series is called the lower normal closure of H in G and will be denoted by HG,∞. Observe that every subgroup H is contranormal in its lower normal closure. In finite groups, the subgroup HG,∞ is called the subnormal closure of H in G. The rationale for this is the following. In a finite group G, the normal closure series of every subgroup H is finite, and HG,∞ is the smallest subnormal subgroup of G containing H. A subgroup H is called descendant in G if H coincides with its lower normal closure HG,∞. An important particular case of descendant subgroups are subnormal sub- groups. A subnormal subgroup is exactly a descending subgroup having finite normal closure series. These subgroups strongly affect structure of a group. For example, it is not hard to prove that if every subgroup of a locally ( soluble-by-finite ) group is descendant, then this group is locally nilpotent. If every subgroup of a group G is subnormal, then, by a remarkable result due to W. Möhres [75], G is soluble. Subnormal sub- groups have been studied very thoroughly for quite a long period of time. We are not going to consider this topic here since it has been excellently presented in the survey of C. Casolo [19]. However, we need to admit that, with the exception of subnormal subgroups, we have no significant information regarding descendant subgroups. The next results connect the conditions of generalized nilpotency to descendant subgroups. 4.4. (L.A. Kurdachenko, I.Ya. Subbotin [61]) Let G be a group, every subgroup of which is descendant. If G is FC–hypercentral, then G is hypercentral. 4.5. (L.A. Kurdachenko, I.Ya. Subbotin [62]) Let G be a generalized minimax group. Then every subgroup of G is descendant if and only if G is nilpotent. If every subgroup of a group G is descendant, then G does not include proper contranormal subgroups. The study of groups without contranor- mal subgroups is the next logical step. We observe that every non-normal maximal subgroup of an arbitrary group is contranormal. Since a finite group whose maximal subgroups are normal is nilpotent, we come to the following criterion of nilpotency of finite groups in terms of contranormal subgroups: A finite group G is nilpotent if and only if G does not include proper contranormal subgroups. The question on existing of an analog of this criterion for infinite groups is very natural. However, in general, the absence of contranormal subgroups does not imply nilpotency. In fact, there exist non-nilpotent groups all subgroups of which are subnormal. The first such example has Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 61 been constructed by H. Heineken and I.J. Mohamed [41]. Nevertheless, for some classes of infinite groups the absence of contranormal subgroups does imply nilpotency of a group. The groups without proper contra- normal subgroups have been considered in papers [51, 52]. We show the main results of these articles. 4.6. Let G be group and H be a normal soluble-by-finite subgroup such that the factor-group G/H is nilpotent. Suppose that H satisfies Min−G. If G has no proper contranormal subgroups, then G is nilpotent. In particular, if soluble-by-finite group G without proper contranormal subgroups satisfies the minimal condition on normal subgroups, then G is nilpotent. 4.7. Let G be a group and H be a normal Chernikov subgroup. Sup- pose that G/H is nilpotent. If G has no proper contranormal subgroups, then G is nilpotent. In particular, a Chernikov group without proper con- tranormal subgroups is nilpotent. 4.8. Let G be group and C be a normal subgroup such that the factor-group G/C is nilpotent. Suppose that C has a finite series of G−invariant subgroups 〈1〉 = C0 ≤ C1 ≤ ... ≤ Cn = C whose factors Cj/Cj−1, 1 ≤ j ≤ n, satisfy one of the following conditions: (i) Cj/Cj−1 is finite; (ii) Cj/Cj−1 is hyperabelian and minimax; (iii) Cj/Cj−1 is hyperabelian and finitely generated; (iv) Cj/Cj−1 is abelian and satisfies Min−G. If G has no proper contranormal subgroups, then G is nilpotent. Let G be a group and let A be an infinite normal abelian subgroup of G. We say that A is a G−quasifinite subgroup if every proper G−invariant subgroup of A is finite. This means that either A includes a proper finite G−invariant subgroup B such that A/B is G−simple, or A is an union of all finite proper G−invariant subgroups. 4.9. Suppose that a group G includes a normal subgroup C such that the factor-group G/C is nilpotent. Suppose that C has a finite series of G−invariant subgroups 〈1〉 = C0 ≤ C1 ≤ ... ≤ Cn = C whose factors Cj/Cj−1, 1 ≤ j ≤ n, satisfy one of the following conditions: (i) Cj/Cj−1 is finite; (ii) Cj/Cj−1 is hyperabelian and minimax; Jo u rn al A lg eb ra D is cr et e M at h .62 On some developments in investigation of groups (iii) Cj/Cj−1 is hyperabelian and finitely generated; (iv) Cj/Cj−1 is abelian and G−quasifinite. If G has no proper contranormal subgroups, then G is nilpotent. The following useful assertions are almost direct consequences of this theorem 4.10. Let G be a group and let C be a normal soluble subgroup of G such that G/C is nilpotent. Suppose that C has a finite G−chief series. If G has no proper contranormal subgroups, then G is nilpotent. 4.11. Let G be a group and let C be a normal soluble subgroup of G such that G/C is nilpotent. Suppose that C is a hyperabelian minimax subgroup. If G has no proper contranormal subgroups, then G is nilpo- tent. In particular, if G is hyperabelian minimax group without proper contranormal subgroups, then G is nilpotent. 4.12. Let G be a group and let C be a normal soluble subgroup of G such that G/C is nilpotent. Suppose that C is a Chernikov subgroup. If G has no proper contranormal subgroups, then G is nilpotent. In particular, if G is a Chernikov group without proper contranormal subgroups, then G is nilpotent. 4.13. Let G be a group and let C be a normal soluble subgroup of G such that G/C is nilpotent. Suppose that C is a hyperabelian finitely generated subgroup. If G has no proper contranormal subgroups, then G is nilpotent. In particular, if G is hyperabelian finitely generated group without proper contranormal subgroups, then G is nilpotent. 4.14. Suppose that the group G includes a normal G−minimax sub- group C such that G/C is a nilpotent group of finite section rank. If G has no proper contranormal subgroups, then G is nilpotent. 5. Transitivity of generalized normality We mentioned above some important results on transitivity of normal- ity. Transitivity of such important subgroup properties as pronormality, abnormality and other related to them properties have been studied by L.A. Kurdachenko, I.Ya. Subbotin, and J.Otal (see, [60], [59], and [49]). The groups, in which pronormality is transitive are called TP -groups and the groups in which all subgroups are TP−groups are called T̄P−groups. The following descriptions of soluble T̄P−groups and TP−groups have been obtained in [60] 5.1. (L.A. Kurdachenko, I.Ya. Subbotin [60]) Let G be a locally soluble group. Then G is a T̄P−group if and only if G is a T̄−group. Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 63 5.2. (L.A. Kurdachenko, I.Ya. Subbotin [60]) Let G be a periodic soluble group. Then G is a TP -group if and only if G = A ⋋ (B × P ) where (i) A and B are abelian 2́-subgroups in G and P is a 2-subgroup (if P is non-identity); (ii) π(A) ∩ π(A) = ∅; (iii) P is a T -group; (iv) [G,G] = A× [P, P ]; (v) every subgroup of [G,G] is G−invariant; (vi) A is a complement to every Sylow π(B × P )−subgroup of G. In [60] the authors list all types of periodic soluble TP−groups. The following theorem completes a description of soluble TP−groups. 5.3. (L.A. Kurdachenko, I.Ya. Subbotin [60]) Let G be a non-periodic soluble group. (i) If CG([G,G]) is non-periodic, then G is a TP− group if and only if G is a T−group. (ii) If CG([G,G]) is periodic, then G is a TP−group if and only if G is a hypercentral T−group. In this setting, it is interesting to mention the following, most general yet, result on transitivity of abnormal subgroups. 5.4. (L.A. Kurdachenko, I.Ya. Subbotin [62]) Let G be a group and suppose that A is a normal subgroup of G such that G/A has no proper abnormal subgroups. If A satisfies the normalizer condition, then abnormality is transitive in G. In particular, if G is metanilpotent group, then abnormality is tran- sitive in G. Recall the following interesting property of pronormal subgroups: Let G be a group, H,K be the subgroups of G and H ≤ K. If H is a subnormal and pronormal subgroup in K, then H is normal in K. We say that a subgroup H of a group G is transitively normal if H is normal in every subgroup K ≥ H in which H is subnormal [63]. In [78], these subgroups have been introduced under a different name. Namely, a subgroup H of a group G is said to satisfy the subnormalizer condition in G if for every subgroup K such that H is normal in K we have NG(K) ≤ NG(H). We say that a subgroup H of a group G is strong transitively normal, if HA/A is transitively normal for every normal subgroup A of the group G Jo u rn al A lg eb ra D is cr et e M at h .64 On some developments in investigation of groups [63]. Since the homomorphic image of pronormal subgroup is pronormal, we can conclude that every pronormal subgroup is a strong transitively normal subgroup. 5.5. (L.A. Kurdachenko, I.Ya. Subbotin [63]). Let G be a group, H be a hypercentral subgroup of G. Suppose that G includes a normal soluble subgroup R such that G/R is hypercentral. If H is strong transitively normal in G and R satisfies Min−H, then H is a pronormal subgroup of G. As direct corollaries we can mention the following results [63]. 5.6. Let G be a group, H be a hypercentral subgroup of G. Suppose that G includes a normal soluble Chernikov subgroup R such that G/R is hypercentral. If H is strong transitively normal in G, then H is a pronormal subgroup of G. In particular, if G is a soluble Chernikov group and H is a hypercentral strong transitively normal subgroup of G, then H is pronormal in G. 5.7. Let G be a group, H be a hypercentral subgroup of G. Suppose that G includes a normal soluble subgroup R such that G/R is hyper- central. If H is a polynormal subgroup of G and R satisfies Min −H (in particular, if R is Chernikov), then H is pronormal in G. The following theorem from [78] directly follows from this. 5.8. Let G be a soluble finite group, H be a nilpotent subgroup of G. If H is a polynormal subgroup of G, then H is a pronormal subgroup of G. A subgroup H is said to be paranormal in a group G if H is contra- normal in 〈H,Hg〉 for all elements g ∈ G ( M.S. Ba and Z.I. Borevich [2]). Every pronormal subgroup is paranormal, and every paranormal subgroup is polynormal [2]. 5.9. (L.A. Kurdachenko, I.Ya. Subbotin [63]). Let G be a group, H be a hypercentral subgroup of G. Suppose that G includes a normal soluble subgroup R such that G/R is hypercentral. If H is a paranormal subgroup of G and R satisfies Min−H ( in particular, if R is a Chernikov group), then H is pronormal in G. As a corollary we obtain 5.10. Let G be a soluble finite group, H be a nilpotent subgroup of G. If H is a paranormal subgroup of G, then H is a pronormal subgroup of G. In [82], the following criterion of pronormality of a nilpotent subgroup in a finite group has been established. Jo u rn al A lg eb ra D is cr et e M at h .V. V. Kirichenko, L. A. Kurdachenko 65 5.11. Let G be a nilpotent-by-abelian finite group, H be a nilpotent subgroup of G. If H is transitively normal in G, then H is a pronormal subgroup of G. The article [63] contains the following useful strong generalization of this criterion on some infinite cases. 5.12.(L.A. Kurdachenko, I.Ya. Subbotin [63]) Let G be a group, H be a hypercentral subgroup of G. Suppose that G includes a normal nilpotent subgroup R such that G/R is hypercentral. If H is transitively normal in G and R satisfies Min−H ( in particular, if R is Chernikov), then H is a pronormal subgroup of G. As a corollary we obtain 5.13. Let G be a nilpotent-by-hypercentral Chernikov group, H be a hypercentral subgroup of G. If H is transitively normal in G, then H is a pronormal subgroup of G. A subgroup H of a group G is called weakly normal if Hg ≤ NG(H) implies that g ∈ NG(H) (K.H. Müller [77]). We note that every pronor- mal subgroup is weakly normal [3], every weakly normal subgroup satisfies the subnormalizer condition [3], and hence it is transitively normal in G. Thus from above result we obtain 5.14. (L.A. Kurdachenko, I.Ya. Subbotin [63] Let G be a group, H be a hypercentral subgroup of G. Suppose that G includes a normal nilpotent subgroup R such that G/R is hypercentral. If H is weakly normal in G and R satisfies Min − H ( in particular, if R is a Chernikov group), then H is a pronormal subgroup of G. A subgroup H of a group G is called an H– subgroup if NG(H)∩Hg ≤ H for all elements g ∈ G [6]. Note that every H– subgroup is transitively normal [6]. Therefore, from above result we obtain 5.15 (L.A. Kurdachenko, I.Ya. Subbotin [63]) Let G be a group, H be a hypercentral subgroup of G. Suppose that G includes a normal nilpotent subgroup R such that G/R is hypercentral. If H is an H– subgroup of G and R satisfies Min−H ( in particular, if R is a Chernikov group), then H is a pronormal subgroup of G. Some properties of transitively normal subgroups (under another name) have been considered in the paper [29], which in particular, contains the following result. 5.16. Let G be an FC−group, H be a transitively normal subgroup of G. If H is a p−subgroup for some prime p, then H is a pronormal subgroup of G. 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Kirichenko Department of Mechanics & Mathematics, Kyiv National Taras Shevchenko University, Volodymyrska, 64, Kyiv, 01033, Ukraine E-Mail: vkir@univ.kiev.ua L. A. Kurdachenko Department of Algebra, School of Mathe- matics and Mechanics, National University of Dnepropetrovsk, Gagarin Prospect 72, Dnepropetrovsk 10, 49010, Ukraine E-Mail: lkurdachenko@i.ua Received by the editors: 05.02.2010 and in final form 02.03.2010.

On some developments in investigation of groups with prescribed properties of generalized normal subgroups

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