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 maximum condition for nonabelian subgroups

We study locally soluble-by-finite groups with the maximum condition for nonabelian subgroups. These groups do not necessarily satisfy the maximum condition for all subgroups. But they are finitely generated and metabelian-byfinite.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 925–930, July–August, 1991.     

Locally supersolvable extensions of Abelian groups

An extension E of an Abelian group A by a locally supersolvable group is studied under the assumption that A has a finite series of subgroups normal in E, each of whose factors satisfies one of the following conditions: 1) min-E; 2) max-E; 3) the factor has finite rank and is torsion-free. It is proved that if E induces in the factors of the series in A hypercyclic groups of automorphisms and A is the locally supersolvable coradical of E, then A is complemented in E and all complements are conjugate in E. An analog of this result for locally nilpotent extensions of Abelian groups is also noted.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 908–912, July, 1990.     

Infinite nonabelian groups with minimal condition for non-normal subgroups

Infinite nonabelian groups which do not possess a descending chain of non-normal subgroups are examined. It is established that under certain additional conditions, in particular the condition of the existence of a normal system with finite factors, the class of all such groups consists only of infinite extremal nonabelian groups and infinite hamiltonian groups, see [6].Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 11–18, July, 1969.     

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.     

The exponents of nonabelian tensor products of groups

We prove that the exponent of the nonabelian tensor product of two locally finite groups can be bounded in terms of exponents of given groups. Several estimates for the exponents of nonabelian tensor squares are obtained. In particular, if the group G is nilpotent of class ≤3 and of finite exponent, then the exponent of its nonabelian tensor square divides the exponent of G.     

A certain class of metabelian groups with complemented subgroups

The metabelian periodic groups, in which all serving subgroups of each maximal abelian subgroup are complemented, are studied. A presentation of arbitrary groups of this kind is given.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1093–1097, July–August, 1991.     

2-Groups with normal noncyclic subgroups

We study nearly-hamiltonian 2-groups (¯H₂-groups). A nonabelian 2-group having at least one proper noncyclic subgroup is called an ¯H₂-group if all noncyclic subgroups are normal.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 75–84, July, 1968.The author expresses his deep gratitude towards Professor S. N. Chernikov for guidance while writing this paper.     

Abelian limits of compact groups in the space of closed subgroups

We show that if an Abelian subgroup is the limit of compact groups in the space of closed subgroups of a locally compact group in the Vietoris topology, then it is inductively compact. The results may be applied to study normality and metrizability of the space of closed subgroups of a Lie group, the limits of monothetic groups, and also other issues related to the theory of the space of closed subgroups.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 137–141, 1988.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

Topological groups with compact proper invariant subgroups

A construction is given which lets one construct examples of locally compact inductively pronilpotent groups for which all proper invariant subgroups are compact and which do not contain proper open invariant subgroups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 708–710, May, 1990.     

Topological groups with the generalized minimality condition for subgroups

Locally compact, locally solvable groups with the generalized minimality condition for closed Abelian subgroups are studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1394–1402, October, 1990.     

Solvable infinite-dimensional linear groups with restrictions on the nonabelian subgroups of infinite rank

Under study are the solvable nonabelian linear groups of infinite central dimension and sectional p-rank, p ≥ 0, in which all proper nonabelian subgroups of infinite sectional p-rank have finite central dimension. We describe the structure of the groups of this class.     

On the Conjugacy problem for Carter subgroups

We prove that the Carter subgroups of a finite group are conjugate if so are the Carter subgroups of the group of induced automorphisms for every nonabelian composition factor.     

Bundles, Handcuffs, and Local Freedom

We answer a question of J. Anderson's by producing infinitely many commensurability classes of fibered hyperbolic 3-manifolds whose fundamental groups contain subgroups that are locally free and not free. These manifolds are obtained by performing 0–surgery on a collection of knots with the same properties.     

Groups with finitely many derived subgroups of non-normal subgroups

A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite derived subgroup. This result is generalized here, by proving that every locally graded group with finitely many derived subgroups of non-normal subgroups has finite derived subgroup. Moreover, locally graded groups having only finitely many derived subgroups of infinite non-normal subgroups are completely described. Received: 25 April 2005     

Groups with the minimality condition for infinitely generated subgroups

We establish that, in the class of locally almost solvable groups, the minimality condition for infinitely generated subgroups is equivalent to the minimality condition for (all) subgroups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1280–1282, September, 1994.     

Finite 2-groups all of whose nonabelian subgroups are generated with involutions

In this note we determine finite nonabelian 2-groups G all of whose nonabelian subgroups are generated by involutions and show that such groups must be quasi-dihedral. This solves the problem Nr. 1595 for p = 2 in [1].