Groups with Polycyclic-by-Finite Conjugate Classes of Subgroups

Kurdachenko, Leonid A.; Otal, Javier; Soules, Panagiotis

doi:10.1081/agb-200036758

Cited by 2 publications

(3 citation statements)

References 8 publications

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“…Theorem 2.3 [6]. A group G has polycyclic-by-finite classes of a conjugate subgroups if and only if it is central-by-(polycyclic-by-finite).…”

Section: Francesco Russomentioning

confidence: 99%

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Anti-CC-Groups and Anti-PC-Groups

Russo

2007

International Journal of Mathematics and Mathematical Sciences

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A groupGhas Černikov classes of conjugate subgroups if the quotient groupG/coreG(NG(H))is a Černikov group for each subgroupHofG. An anti-CCgroupGis a group in which each nonfinitely generated subgroupKhas the quotient groupG/coreG(NG(K))which is a Černikov group. Analogously, a groupGhas polycyclic-by-finite classes of conjugate subgroups if the quotient groupG/coreG(NG(H))is a polycyclic-by-finite group for each subgroupHofG. An anti-PCgroupGis a group in which each nonfinitely generated subgroupKhas the quotient groupG/coreG(NG(K))which is a polycyclic-by-finite group. Anti-CCgroups and anti-PCgroups are the subject of the present article.

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“…Theorem 2.3 [6]. A group G has polycyclic-by-finite classes of a conjugate subgroups if and only if it is central-by-(polycyclic-by-finite).…”

Section: Francesco Russomentioning

confidence: 99%

“…Each anti-FC-group is an anti-CC-group as testified by definitions. Examples of anti-FC-groups can be found in [13, page 44, lines [1][2][3][4][5][6][7][8][9][10][11][12][13] or [13,Example 3.12]. Of course, each anti-FC-group is an anti-PC-group.…”

Section: Examplesmentioning

confidence: 99%

Anti-CC-Groups and Anti-PC-Groups

Russo

2007

International Journal of Mathematics and Mathematical Sciences

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“…If X = P, then we arrive at the class of PC-groups or the class of all groups with almost polycyclic classes of conjugation. The investigation of this class has been recently originated in [10,11]. In the present paper, we consider a subclass of the class of PC-groups obtained as a result of the generalization of the notion of nearly normal subgroups presented in what follows.…”

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confidence: 98%

On some generalizations of nearly normal subgroups

Semko

Pyskun

2009

Ukr Math J

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Subgroups of this type were introduced by Neumann in [1]. In the cited work, he described the groups any subgroup of which is nearly normal. These groups have finite commutants and, in particular, they are FC-groups. The study of the influence of the properties of nearly normal subgroups on the structure of the group was continued by other researchers. Thus, the groups for which the system of all nearly normal subgroups is dense are investigated in [2]. The groups for which the system of all subgroups that are not nearly normal satisfies the minimality condition are studied in [3]. The groups for which the same system of subgroups satisfies the maximality condition are analyzed in [4]. The groups each subgroup of which is either nearly normal or subnormal are considered in [5]. We also mention the work [6] devoted to the investigation of some properties of the lattice of all nearly normal subgroups.A generalization of the notion of nearly normal subgroups presented in what follows was introduced in [7]. First, we recall several definitions required for our subsequent presentation (see [8], Chap. 3).Let X a class of groups. We say that a group G has X-classes of conjugate elements or G is an X Cgroup if the quotient group G G g G G / ( ) belongs to the class X for each element g of the group G. Here, g G denotes the class of all elements conjugated to the element g, i.e., the subset g xIf X = I is the class of all identity groups, then the class of all IC-groups coincides with the class A of all Abelian groups. Hence, for the proper choice of the class X, the class of XC-groups can be regarded as a natural generalization of the class of Abelian groups.Thus, if X = F is the class of all finite groups, then the class of all FC-groups is exactly the class of all FC-groups or groups with finite classes of conjugate elements. This class is a fairly convenient extension of both the class of all Abelian groups and the class of all finite groups and inherits various properties of these two classes. For this reason, the theory of FC-groups is one of the most developed among the theories of infinite groups.The class C of all Chernikov groups and the class P of all almost polycyclic groups are natural extensions of the class of finite groups. Hence, if X = C, then the class of all CC-groups is exactly the class of all groups with Chernikov classes of conjugate elements introduced by Polovitskii in [9]. If X = P, then we arrive at the

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Groups with Polycyclic-by-Finite Conjugate Classes of Subgroups

Cited by 2 publications

References 8 publications

Anti-CC-Groups and Anti-PC-Groups

Anti-CC-Groups and Anti-PC-Groups

On some generalizations of nearly normal subgroups

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