Modules over nilpotent groups of finite rank

Zaitsev, D. I.; Kurdachenko, Leonid A.; Tushev, A. V.

doi:10.1007/bf01978845

Cited by 19 publications

(7 citation statements)

References 8 publications

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“…Under the conditions of the theorem, L is a torsion-free group. It follows from Corollary 1 of Lemma 2.6 in [18] that its center ζ( ) L does not have a finite system of generating elements. In the subgroup ζ( ) L , we choose a serving G-invariant infinitely generated subgroup C of the least possible 0-rank.…”

Section: Theorem 3 Let G Be a Nonperiodic Almost Solvable Minimax Grmentioning

confidence: 95%

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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Section: Theorem 3 Let G Be a Nonperiodic Almost Solvable Minimax Grmentioning

confidence: 95%

On some generalizations of nearly normal subgroups

Semko

Pyskun

2009

Ukr Math J

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“…Let S/B denote the torsion subgroup of H/B. It follows by the argument given in the first paragraphs of the proof of [14,Theorem 3] that H/S is minimax and we deduce that S/B is not minimax. Since S/B satisfies min-∞-g , it is clear that there are only finitely many primes p such that the p-component of S/B is nontrivial.…”

Section: Proof Of Theorem A(ii)mentioning

confidence: 95%

Locally Soluble-by-Finite Groups with the Weak Minimal Condition on Non-Nilpotent Subgroups

2002

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“…Goretsky [153], L.A. Kurdachenko and H. Smith [165,166,167,168], L.A. Kurdachenko, A.V. Tushev [312]). Zaitsev [175], M.L.…”

Section: Soluble-by-finite) Group Has a Finite Minimax Rank If And Onmentioning

confidence: 99%

“…Robinson showed that every finitely generated hyperabelian group of finite special rank (respectively finite section rank) is minimax [228,233,238,243] (see also D.I. Tushev [312], D. Segal [256,257], D.I. Properties of subnormal and permutable subgroups in minimax groups have been considered by D.C. Brewster and J.C. Lennox [24], D.J.…”

Section: Soluble-by-finite) Group Has a Finite Minimax Rank If And Onmentioning

confidence: 99%