On the theory of groups with generalized minimality condition for closed subgroups

We prove that a topological Abelian locally compact group with generalized minimality condition for closed subgroups is a group of one of the following types: 1) a group with minimality condition for closed subgroups, 2) an additive group of theJ _p -ring of integerp-adic numbers, 3) an additive groupR _p of the field ofp-adic numbers (p is a prime number). Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 3, pp. 398–409, March, 1999.     

Linear groups with minimality condition for some infinite-dimensional subgroups

Let F be a field, let A be a vector space over F, and let GL(F, A) be the group of all automorphisms of the space A. If H is a subgroup of GL(F, A), then we set aug dim_F (H) = dim_F (A(ωFH)), where ωFH is the augmentation ideal of the group ring FH. The number aug dim_F (H) is called the augmentation dimension of the subgroup H. In the present paper, we study locally solvable linear groups with minimality condition for subgroups of infinite augmentation dimension. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1476–1489, November, 2005.     

Periodic groups with the primary minimality condition for certain systems of subgroups

Krasnoyarsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 31, No. 4, pp. 160–165, July–August, 1990.     

Topological{FC}-solvable groups with maximality and minimality conditions for the subgroups

Topological groups having a normal series, all of whose factors are -groups, are studied.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 393–400, October, 1969.The author would like to thank his scientific director, V. I. Ushakov.     

On topological groups with weak minimality (maximality) condition for noncompact subgroups

We present the structure of a locally compact solvablep-group satisfying the weak minimality (maximality) condition for noncompact subgroups. As a consequence, we obtain the structure of a locally compact prosolvablep-group satisfying the minimality (maximality) condition for noncompact sub-groups. We also construct an example to demonstrate that these results are not true for arbitrary inductively compact locally compact totally disconnected solvable groups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1393–1398, October, 1994.     

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.     

On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups

We characterize groups without nontrivial perfect sections (in particular, solvable groups) with the minimality condition for the subgroups without hypercentral subgroups of finite index. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 10, pp. 1425–1430, October, 1999.     

Groups satisfying the minimality condition for non-almost-layer-finite subgroups

The following theorem characterizing the class of almost layer-finite groups in the class of periodic groups without involutions is proved: Each conjugate-biprimitively finite group without involutions that satisfies the minimality condition for non-almost-layer-finite subgroups is almost layerfinite.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1002–1008, July–August, 1991.     

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.     

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.     

Groups satisfying the condition of weak minimality for subgroups of derived length two

We consider non-Abelian solvable and radical groups that satisfy the condition of weak minimality for subgroups of derived length two. We show that these groups are minimax groups. This is not true for locally solvable groups. We give an example of a group of derived length three that satisfies the condition of weak minimality for the indicated subgroups but is not a minimax group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp.1203–1207, September, 1994.