Groups with some restrictions on non-Baer subgroups

Ricerche di Matematica - It is proved that if G is an $$\mathfrak {X}$$ -group of infinite rank whose proper subgroups of infinite rank are Baer groups, then so are all proper subgroups of G, where...     

A Sufficient Condition for Mi-groups

In this note, we give a sufficient condition for Mi-group. In particular, we show that if a finite group G is the semidirect product of two subgroups with coprime orders, in which one is a Sylow tower group and its Sylow subgroups are all abelian, and the other is an Mi-group and all of its proper subgroups are also Mi-groups, then G is an Mi-group.     

Finite groups with seminormal Schmidt subgroups

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB₁ is a proper subgroup of G, for every proper subgroup B₁ of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary. Supported by BelFBR grant Nos. F05-341 and F06MS-017. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 448–458, July–August, 2007.     

极小非PNN-群

有限群G称作PNN-群,如果G的每一个极小子群X要么正规于G,要么是自正规化的.本文将讨论PNN-群的性质,并给出极小非PNN-群,也就是本身不是PNN-群,每个真子群都是PNN-群的有限群的完全分类.     

A note on groups of infinite rank with modular subgroup lattice

It is proved that if \(G\) is a (generalized) soluble group of infinite rank in which all proper subgroups of infinite rank are permodular, then the subgroup lattice of \(G\) is permodular. As a consequence of this theorem, we obtain shorter proofs for corresponding known results concerning normal or permutable subgroups of groups of infinite rank.     

Determinazione deiT 2-gruppi finiti

Summary A T-group is a group in which normality is transitive, a T₁-group is a group which is not a T-group but all of whose proper subgroups are T-groups, and a T₂-group is a group which is not a T- group or a T₁-group but all of whose proper subgroups are T-groups or T₁-groups. In this note we determine all the finite T₂-groups, and we show that all T₂-groups which are either soluble or 2-groups are finite. We study also the groups in which every proper soluble subgroup is a T-group or a T₁-group.

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. (C.N.R.).     

一类可解的（q）－群

有限群Ｇ叫（ｑ）－群，如果Ｇ中每个次正规子群均为拟正规子群，群Ｇ叫Ｅｑ－群，若Ｇ中每个子群在Ｇ中拟正规或自正规，有限群Ｇ叫内Ｅｑ－群，如果Ｇ本身不是Ｅｑ－群，但Ｇ的每个真子群是Ｅｑ－群，本文确定了Ｅｑ－群的结构与内Ｅｑ－群的分类．     

能表示成四个真子群并的有限群

本文研究了子群覆盖问题.利用计算子群阶的方法,给出了能表示成p+1或p+2个真子群并的有限群G的结构,这里p是群G阶的最小素因子,从而推广了文献[1]中的结果.     

On fixed subgroups of maximal rank

We show that, in the free group F of rank n, n is the maximal length of strictly ascending chains of maximal rank fixed subgroups, that is, rank n subgroups of the form Fix^{^} for some 4> L Aut(F). We further show that, in the rank two case, if the intersection of an arbitrary family of proper maximal rank fixed subgroups has rank two then all those subgroups are equal. In particular, every G < Aut(F) with r(FixG) = 2 is either trivial or infinite cyclic.     

Linear bounds for the normal covering number of the symmetric and alternating groups

Monatshefte für Mathematik - The normal covering number $$\gamma (G)$$ of a finite, non-cyclic group G is the minimum number of proper subgroups such that each element of G lies in some...     

Groups with the weak minimal condition for non-subnormal subgroups

Let G be a group with the property that there are no infinite descending chains of non-subnormal subgroups of G for which all successive indices are infinite. The main results are as follows. If G is locally nilpotent then either G is minimax or G has all subgroups subnormal; if G is a Baer group then all subgroups of G are subnormal. It is also proved that a generalised radical group with this property is soluble-by-finite and either is minimax or has all subgroups subnormal.     

Groups with minimax factor groups

We consider groups in which every normal subgroup which is not minimax determines a minimax quotient group. If G is a group with this property then it is clear that either G contains an ascending chain of normal subgroups with minimax quotient groups or G contains a normal minimax subgroup H such that G/H does not contain any non-identity normal minimax subgroups. In particular, every proper factor group of G/H is minimax. In the present paper we study the first case.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 620–625, May, 1990.     

On nilpotency of higher commutator subgroups of a finite soluble group

Archiv der Mathematik - Let G be a non-abelian finite simple group. In addition, let $$\Delta _G$$ be the intersection graph of G, whose vertices are the proper non-trivial subgroups of G, with...     

A generalization of Schmidt’s Theorem on groups with all subgroups nilpotent

According to Schmidt’s Theorem a finite group whose proper subgroups are all nilpotent (or a finite group without non-nilpotent proper subgroups) is solvable. In this paper we prove that every finite group with less than 22 non-nilpotent subgroups is solvable and that this estimate is sharp.     

有限群的最大子群的性质对群结构的影响

有限群G的一个子群称为在G中是π-拟正规的若它与G的每一个Sylow-子群是交换的.G的一个子群H称为在G中是c-可补的若存在G的子群N使得G=HN且H∩N≤HG=CoreG(H).本文证明了:设F是一个包含超可解群系u的饱和群系,G有一个正规子群H使得G/H∈F.则G∈F若下列之一成立:(1)H的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的;(2)F*(H)的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的,其中F*(H)是H的广义Fitting子群.此结论统一了一些最近的结果.     

On groups whose all proper subgroups have Chernikov derived subgroups

We study the groups for which the derived subgroups of all proper subgroups are Chernikov subgroups under condition that they possess a normal system whose factors are locally graduated.     

On the normal indices of proper subgroups of finite groups

We extended the normal index from maximal subgroups to proper subgroups. We give a quantitative version of all results obtained by using c-normal subgroups and obtain some new characterizations of solvable, supersolvable and nilpotent groups by the normal indices of proper subgroups.     

On θ*-completions of F-abnormal Subgroups

Let F be a saturated formation of finite groups. Given a proper subgroup H of a group G, a subgroup H of G is called F-normal in G if G/HG belongs to F;otherwise H is said to be F-abnormal in G. In this paper, we investigate the structure of a finite group byθ?-completions of F-abnormal subgroups.     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.