Characterization of groups with generalized chernikov periodic part

A characterization of groups with generalized Chernikov periodic part is obtained first in the class of groups without elements of order 2 and then without this restriction. Two author’s theorems for periodic groups are generalized to mixed groups. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 270–275, February, 2000.     

Characterization of groups with layer-finite periodic parts

We consider groups without involutions, in which the normalizer of an arbitrary finite nontrivial subgroup, invariant with respect to a certain elementa of prime order p, has layer-finite periodic part. Under a quite weak finiteness condition it is proved that such groups also have layer-finite periodic parts.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 942–946, July–August, 1991.     

Sylow 2-subgroups of the periodic groups saturated with finite simple groups

We prove that every 2-subgroup of a periodic group saturated with groups of Lie type over fields of odd characteristics whose Lie ranks are bounded as a whole is Chernikov. In particular, every such group is locally finite.     

The periodic groups saturated by finitely many finite simple groups

Denote by $\mathfrak{M}$ the set whose elements are the simple 3-dimensional unitary groups U ₃(q) and the linear groups L ₃(q) over finite fields. We prove that every periodic group, saturated by the groups of a finite subset of $\mathfrak{M}$ , is finite.     

On the periodic groups saturated by direct products of finite simple groups

We prove the local finiteness of a periodic group G saturated by direct products of an elementary abelian 2-group of fixed order and the simple groups L ₂(q) under condition that G contains an element of order 4.     

Characterization of simple symplectic groups of degree 4 over locally finite fields of characteristic 2 in the class of periodic groups

Suppose that each finite subgroup of even order of a periodic group containing an element of order 2 lies in a subgroup isomorphic to a simple symplectic group of degree 4 over some finite field of characteristic 2. We prove that in that case the group is isomorphic to a simple symplectic group S ₄(Q) over some locally finite field Q of characteristic 2.     

FC groups whose periodic parts can be embedded in direct products of finite groups

In this note are considered FC groups whose periodic parts can be embedded in direct products of finite groups. It is shown that if the periodic part of an FC group G can be embedded in the direct product of its finite factor groups with respect to the normal subgroups of G whose intersection is the trivial subgroup, then G/Z (G) is a subgroup of a direct product of finite groups. It is also shown that if the periodic part of an FC group G is a group without a center, then G can be embedded in a direct product of finite groups without centers and a torsion-free Abelian group. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 9–20, January, 1977. The author is thankful to the referee for making many valuable remarks.     

On the periodic groups saturated by direct products of finite simple groups. II

We continue the study that was started in [1] of periodic groups saturated with direct products of elementary abelian 2-groups and simple linear groups of dimension 2.     

Imbedding of periodic groups in simple periodic groups

It is proved that each periodic group is isomorphic to a subgroup of some simple periodic group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 845–847, June, 1992.     

On mixed Abelian groups with periodic groups of automorphisms

In the paper, sufficient conditions for the splittability of mixed Abelian groups with periodic automorphism groups are established. Classes of mixed splittable Abelian groups with perfect holomorphs are distinguished. Translated fromMaternaticheskie Zametki, Vol. 61, No. 4, pp. 483–493, April, 1997. Translated by A. I. Shtern