Finiteness conditions and factorizations in infinite groups

Kazarin, L. S.; Kurdachenko, Leonid A.

doi:10.1070/rm1992v047n03abeh000894

Cited by 18 publications

(11 citation statements)

References 95 publications

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“…It is also known that torsion-free locally nilpotent groups that satisfy min-∞-are nilpotent [10,Theorem 2.2]. Finally, we draw the reader's attention to [4]. For the most part, this is a survey of the work of Zaicev; it contains many interesting results that are related to our work here.…”

Section: Introductionmentioning

confidence: 92%

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

2002

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Section: Introductionmentioning

confidence: 92%

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

2002

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“…4b, Sec. 3 in [29]). Robinson uses another term (the minimax length of group, Proposition 10.3 in [21]).…”

Section: Groups Whose All Subgroups Of Infinite Minimax Rank Are Almomentioning

confidence: 99%

Groups with Almost Normal Subgroups of Infinite Rank

Semko¹,

Kuchmenko²

2005

Ukr Math J

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We study classes of groups whose subgroups of some infinite ranks are almost normal.Let ν be a certain property of subgroups. This property can be either internal (e.g., ν can be the property of being normal, subnormal, almost normal, permutable, or complementable) and external (in this case, ν is the property of being a subgroup that belongs to a certain class of groups X ). If G is a group, then Σ non-ν ( ) G (respectively, Σ ν ( ) G ) denotes the system of all subgroups G that do not possess the property ν (respectively, possess the property ν ). One the first problems in group theory (which still remains urgent) is the investigation of the influence of the systems Σ ν ( ) G and Σ non-ν ( ) G on the structure of a group for the most important natural properties ν. The classical paper [1] by Dedekind was the first step in this direction. In this paper, Dedekind studied finite groups all subgroups of which are normal, i.e., groups for which the system Σ norm ( ) G coincides with the system of all subgroups or the system Σ non-norm ( ) G is empty. Later, Miller and Moreno [2] studied finite groups all proper subgroups of which are Abelian, i.e., the system Σ ab ( ) G consists of all proper subgroups or Σ non-ab ( ) G = { G }. One should also mention Shmidt's paper [3], where he studied finite groups all proper subgroups of which are nilpotent, i.e., the system Σ nil ( ) G consists of all proper subgroups or Σ non-nil ( ) G = { G }. After these works, both finite and infinite groups for which the system Σ ν ( ) G is "sufficiently large" or the system Σ non-ν ( ) G is "sufficiently small" were thoroughly studied. This direction turned out to be quite interesting and fruitful, and numerous papers and several monographs have been devoted to it. In the present paper, we consider groups with restrictions imposed on the system Σ non-an ( ) G of all subgroups that are not almost normal. A subgroup H of a group G is called almost normal in G if the set cl G H( the class of all subgroups conjugate to H ) is finite.If a subgroup H is normal in G, then cl G H ( ) = { H }, so that almost normal subgroups can be regarded as a natural generalization of normal subgroups. The subgroup H is almost normal in the group G if and only if its normalizer N H G ( ) has a finite index in G (this explains the name of these subgroups). In [4], Neumann characterized groups all subgroups of which are almost normal (i.e., the set Σ non-an ( ) G is empty) as groups with a center of finite index (groups finite over a center). In [5], Eremin generalized this result and proved that groups all Abelian subgroups of which are almost normal have a center of finite index. In [6], Eremin considered groups all infinite subgroups of which are almost normal (i.e., the set Σ non-an ( ) G consists of only finite subgroups). Locally almost solvable groups all infinite subgroups of which are almost normal were described by Semko, Levishchenko, and Kurdachenko in [7]. Later, the investigation of groups for which the system Σ non-an ( ) G is "sufficiently small"...

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“…For a group G, just infinite ZG-modules were first studied in [11] and [5], quasifinite ZG-modules in [16]. For further references the reader is invited to consult the survey [4]. Our first requirement is as follows.…”

Section: Max-g and Min-g Subgroups Of Locally Nilpotent Groupsmentioning

confidence: 99%