Locally graduated groups with complemented infinite nonprimary subgroups

The investigations of infinite groups with certain systems of complemented infinite subgroups, suggested by Chernikov, are continued. It is proved that an infinic locally graduated, nonprimary group with complemented infinite nonprimary subgroups is locally finite and solvable, and all of its nonprimary subgroups have complements if and only if it is not Chernikov.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 839–842, June, 1992.     

Groups with finitely many normalizers of non-abelian subgroups

Abstract A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite commutator subgroup. Here the structure of locally graded groups with finitely many normalizers of (infinite) non-abelian subgroups is investigated, and the above result is extended to this more general situation. Keywords: normalizer subgroup, metahamiltonian group Mathematics Subject Classification (2000): 20F24     

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.     

On the normalizer condition

It is proved that a torsion-free group , in which every infinite proper subgroup is distinct from its normalizer satisfies the normalizer condition, i.e., every proper subgroup is distinct from its normalizer.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 45–50, January, 1968.     

On Complementability of Nonmetacyclic Subgroups in Finite A-Groups

We study groups G that satisfy the following conditions: (i) G is a finite solvable group with nonprimary metacyclic second subgroup and (ii) all Sylow subgroups of the group G are elementary Abelian subgroups. We describe the structure of groups of this type with complementable nonmetacyclic subgroups.     

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.     

Hopficity and Co-Hopficity in Soluble Groups

We show that a soluble group satisfying the minimal condition for its normal subgroups is co-hopfian and that a torsion-free finitely generated soluble group of finite rank is hopfian. The latter property is a consequence of a stronger result: in a minimax soluble group, the kernel of an endomorphism is finite if and only if its image is of finite index in the group.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1335 – 1341, October, 2004.     

On one property of hypercyclic groups

We prove that the condition of periodicity of all normal Abelian subgroups of a group with increasing normal series with cyclic factors such that infinite factors are central is preserved on passing to subgroups of finite index. We present an example which demonstrates that the requirement that all infinite cyclic factors must be central cannot be omitted.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 436–438, March, 1995.     

On the normalizer problem for integral group rings of torsion groups

In this paper, we investigate the normalizer property for the integral group ring of a torsion group. We show that this property holds for locally finite nilpotent groups. A necessary and sufficient condition for this property to hold for any torsion group is also given. This research was supported in part by a research grant from the Natural Sciences and Engineering Research Council of Canada.     

关于具有给定Sylow子群正规化子的有限群Ⅱ

本文在有限可解群中解决了：任意ｍ－秩≤２的子群闭的局部群系具有性质：“如果群Ｇ的非单位Ｓｙｌｏｗ子群的正规化子属于，则群Ｇ也属于的一个充分必要条件．     

On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

We investigate the relationship between the norm N _G() of infinite subgroups of an infinite group G and the structure of this group. We prove that N _G() is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if N _G() is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup N _G() has finite index in G.     

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.     

On infinite groups whose noncyclic norm has a finite index

We study groups in which the intersection of normalizers of all noncyclic subgroups (noncyclic norm) has a finite index. We prove that if the noncyclic norm of an infinite noncyclic group is locally graded and has a finite index in the group, then this group is central-by-finite and its noncyclic norm is a Dedekind group. Sumy Pedagogical Institute, Sumy. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 678–684, May, 1997.     

CC-groups with periodic central factor

A group G is said to be a group with Černikov conjugacy classes or a CC-group if it induces on the normal closure of each one of its elements a group of automorphisms which is a Černikov group, that is, a finite extension of an abelian group satisfying the minimal condition on subgroups. This concept is a natural extension of that an FC-group, that is, a group in which every element has a finite number of conjugates. It is known that if G is an FC-group then the central factor G/Z(G) is periodic. This result does not hold for CC-groups and in this paper we study CC-groups G in which the central factor G/Z(G) is periodic, a finiteness condition which has a deep influence on the structure of the group G. In particular, we characterize those CC-groups as above that are FC-groups by imposing some additional conditions on their structure. This research has been supported by DGICYT (Spain) PS88-0085     

Infinite-dimensional linear groups with restrictions on subgroups that are not soluble <Emphasis Type="Italic">A</Emphasis><Subscript>3</Subscript>-groups

We are concerned with infinite-dimensional locally soluble linear groups of infinite central dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite central dimension. The structure of groups in this class is described. The case of infinite-dimensional locally nilpotent linear groups satisfying the specified conditions is treated separately. A similar problem is solved for infinite-dimensional locally soluble linear groups of infinite fundamental dimension that are not soluble A₃-groups and all of whose proper subgroups, which are not soluble A₃-groups, have finite fundamental dimension. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 548–559, September–October, 2007.     

On stabilization of Ek chains

We study special subgroups of infinite groups that generalize double centralizers. We analyze sufficient conditions for descending chains of such subgroups to stop after finitely many steps. We discuss whether this phenomenon can happen in the class of groups satisfying chain condition on centralizers.