Yaroslav P. Sysak

On the upper central series of infinite groups

Two relevant theorems by R. Baer and P. Hall show that a group is finite over a term with finite ordinal type of its upper central series if and only if it is finite-by-nilpotent. Extending these results, we prove here that if is any group, the hypercentre factor group is finite if and only if contains a finite normal subgroup such that is hypercentral (where the hypercentre of is defined as the last term of its upper central series).

On locally finite groups factorized by locally nilpotent subgroups

Abstract The structure of a periodic radical group G = AB , factorized by two locally nilpotent subgroups A and B , is investigated. In particular, many theorems already known for finite produts of nilpotent groups are extended to the case of periodic radical groups.

On products of groups with abelian subgroups of small index

Abstract It is proved that every group of the form G = A ⁢ B {G=AB} with two subgroups A and B each of which is either abelian or has a quasicyclic subgroup of index 2 is soluble of derived length at most 3. In particular, if A is abelian and B is a locally quaternion group, this gives a positive answer to Question 18.95 of the “Kourovka notebook” posed by A. I. Sozutov.