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 Influence of Contranormal Subgroups on the Structure of Infinite Groups

Following Rose, a subgroup H of a group G is called contranormal, if G = H ^G . In certain sense, contranormal subgroups are antipodes to subnormal subgroups. It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups. However, for the infinite groups this criterion is not valid. There are examples of non-nilpotent infinite groups whose subgroups are subnormal; in paricular, these groups have no contranormal subgroups. Nevertheless, for some classes of infinite groups, the absence of contranormal subgroups implies the nilpotency of the group. The current article is devoted to the search of such classes. Some new criteria of nilpotency in certain classes of infinite groups have been established.     

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.     

Locally nilpotent groups satisfying the weak minimum or maximum condition for subgroups of bounded nilpotency class

We study locally nilpotent groups containing subgroups of classc, c>1, and satisfying the weak maximum condition or the weak minimum condition on c-nilpotent subgroups. It is proved that nilpotent groups of this type are minimax and periodic locally nilpotent groups of this type are Chernikov groups. It is also proved that if a group G is either nilpotent or periodic locally nilpotent and if all of its c-nilpotent subgroups are of finite rank, then G is of finite rank. If G is a non-periodic locally nilpotent group, these results, in general, are not valid.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 384–389, March, 1992.     

Coradicals of subnormal subgroups

IfF is a nonempty formation, then theF-coradical of a finite group G is the intersection of all those normal subgroups N of G for which G / N F. We study the structure of theF-coradical of a group generated by two subnormal subgroups of a finite group. The results are used to reveal properties sufficient for theF-coradicals of subnormal subgroups to be permutable, forF a composition formation.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 493–513, September-October, 1995.     

Groups whose Proper Subgroups of Infinite Rank Have a Transitive Normality Relation

A group G is called a T-group if all its subnormal subgroups are normal, and G is a ${\bar{T}}$ -group if every subgroup of G has the property T. It is proved here that if G is a locally soluble group whose proper subgroups of infinite rank have the T-property, then either G is a ${\bar{T}}$ -group or it has finite rank.     

Groups whose all subgroups are ascendant or self-normalizing

This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972, 125(1), 1–16].     

Groups with small deviation for non-subnormal subgroups

We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at most 1 is nilpotent, while a Baer group with deviation at most 1 has all of its subgroups subnormal.      

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．     

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.     

On modules over integer-valued group rings of locally soluble groups with rank restrictions imposed on subgroups

We study a \mathbbZG \mathbb{Z}G -module A such that \mathbbZ \mathbb{Z} is the ring of integer numbers, the group G has an infinite sectional p-rank (or an infinite 0-rank), C _G (A) = 1, A is not a minimax \mathbbZ \mathbb{Z} -module, and, for any proper subgroup H of infinite sectional p-rank (or infinite 0-rank, respectively), the quotient module A/C _A (H) is a minimax \mathbbZ \mathbb{Z} -module. It is shown that if the group G is locally soluble, then it is soluble. Some properties of soluble groups of this kind are discussed.     

The semiprimitivity problem for group algebras of locally finite groups

LetK[G] be the group algebra of a locally finite groupG over a fieldK of characteristicp>0. IfG has a locally subnormal subgroup of order divisible byp, then it is easy to see that the Jacobson radical ?K[G] is not zero. Here, we come close to a complete converse by showing that ifG has no nonidentity locally subnormal subgroups, thenK[G] is semiprimitive. The proof of this theorem uses the much earlier semiprimitivity results on locally finite, locallyp-solvable groups, and the more recent results on locally finite, infinite simple groups. In addition, it uses the beautiful properties of finitary permutation groups.     

Isomorfismi tra le strutture subnormali dei gruppi

Summary In this paper order-isomorphisms between subnormal structures of subsoluble groups are considered, and the images of generalized nilpotent groups in such isomorphisms are studied. A result of Pazderski about the Fitting subgroup of finite soluble groups is also extended to the upper Baer series of subsoluble groups, and an extension to infinite groups of a theorem of Heineken about isomorphisms between lattices of subnormal subgroups of finite groups is given.

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

Torsionsgruppen,deren Untergruppen alle subnormal sind

We prove that a group, which is the extension of a nilpotent torsion group by a soluble group of finite exponent and all of whose subgroups are subnormal, is nilpotent. The problem can be easily reduced to the investigation of extensions of abelian torsion groups by elementary abelian p-groups with all subgroups of these extensions subnormal.     

关于几乎可解群

本文推广了关于局部有限群的Asar定理及p.Hall—Kulatilaka,Kargapolov定理.     

Cuccia定理的一个推广

For a finite group G,let S(G)be the set of minimal subgroups of odd order of G which are complemented in G.It is proved that if every minimal subgroup X of odd order of G which does not belong to S(G),C_G(X)is either subnormal or abnormal in G.Then G solvable.     

Some classes of groups with the weak minimality and maximality conditions for normal subgroups

This paper deals with groups satisfying the weak minimality (maximality) condition for normal subgroups and having an ascending series of normal subgroups whose factors are finite or Abelian of finite rank. It is proved that if G is such a group, then it contains a periodic hypercentral normal subgroup H satisfying the Min-G condition such that G/H is minimax and almost solvable.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 8, pp. 1050–1056, August, 1990.     

Groups Whose Finite Homomorphic Images are Metahamiltonian

A group G is metahamiltonian if all its non-abelian subgroups are normal. It is proved here that a finitely generated soluble group is metahamiltonian if and only if all its finite homomorphic images are metahamiltonian; the behaviour of soluble minimax groups with metahamiltonian finite homomorphic images is also investigated. Moreover, groups satisfying the minimal condition on non-metahamiltonian subgroups are described.     

Plancherel identity for semisimple hopf algebras

In this paper we discuss the structure of some product G =AB of nilpotent subgroups A and B. In particular we prove that if G is a minimax soluble group or a finitely generated linear group and if it does not have non-trivial periodic normal subgroups, then G is metanilpotent.     

Residually Finite Groups with all Subgroups Subnormal

The following result is established. THEOREM. Let G be a periodic, residually finite group with allsubgroups sub-normal. Then G is nilpotent. The well-known groups of Heineken and Mohamed [1] show thatthe hypothesis of residual finiteness cannot be omitted here,while examples in [5] show that a residually finite group withall subgroups subnormal need not be nilpotent. The proof ofthe Theorem will use the results of Möhres that a groupwith all subgroups subnormal is soluble [3] and that a periodichypercentral group with all subgroups subnormal is nilpotent[4]. Borrowing an idea from [2], the plan is to construct certainsubgroups H and K that intersect trivially, and to show thatthe subnormality of both leads to a contradiction. 1991 MathematicsSubject Classification 20E15.