Groups satisfying the weak minimal condition

Zaitsev, D. I.

doi:10.1007/bf01085210

Cited by 14 publications

(8 citation statements)

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“…Since G satisfies Max-ig-n-, there exists an infinite subset P,i of Ai such that X~i <1 G, for i = 1 and 2. Therefore X = X~ [2] or [13]) apply to show that G is either minimax or a Dedekind group.…”

Section: A##eamentioning

confidence: 96%

Groups with a maximality condition for some non-normal subgroups

Cutolo

Kurdachenko²

1995

Geom Dedicata

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Section: A##eamentioning

confidence: 96%

Groups with a maximality condition for some non-normal subgroups

Cutolo

Kurdachenko²

1995

Geom Dedicata

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“…A locally almost solvable group G has a finite minimax rank if and only if it is minimax, i.e., the group G has a finite subnormal series each factor of each satisfies the Max or Min condition (Zaitsev [30]). Note that, for locally solvable groups, this theorem is proved in [31]. In addition, if all Abelian subgroups of a radical group G have finite minimax ranks, then the group G is minimax (Baer [32] and Zaitsev [33] [18]).…”

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

confidence: 97%

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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“…The weak conditions have been introduced in Baer (1968) and Zaitsev (1968). We recall their definitions in the most general way.…”

Section: Muñoz-escolano Et Almentioning

confidence: 99%

Periodic Linear Groups with the Weak Chain Conditions on Subgroups of Infinite Central Dimension

Escolano

Otal

Semko³

2008

Communications in Algebra

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Let V be a vector space over a field F . If G ≤ GL V F , the central dimension of G is the F -dimension of the vector space V/C V G . In Dixon et al. (2004) and Kurdachenko and Subbotin (2006), soluble linear groups in which the set icd G of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. In this article we study periodic locally radical linear groups, in which the set icd G satisfies one of the weak chain conditions: the weak minimal condition or the weak maximal condition.

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Groups satisfying the weak minimal condition

Cited by 14 publications

References 1 publication

Groups with a maximality condition for some non-normal subgroups

Groups with a maximality condition for some non-normal subgroups

Groups with Almost Normal Subgroups of Infinite Rank

Periodic Linear Groups with the Weak Chain Conditions on Subgroups of Infinite Central Dimension

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