On the theory of groups with extremal layers

Abstract

If m is a positive integer or co, the m layer of a group is the subgroup generated by all the elements of order m. Following Cemikov we call a group extremal if it is a finite extension of an abelian group satisfying the minimal condition for subgroups: extremal groups turn up frequently in the theory of infinite groups and it is an unsolved problem whether every group satisfying the minimal condition is extremal. In this paper we study groups all of whose layers are extremal or EL groups for short. It is clear that an EL group is countable and locally finite since extremal groups have these properties and since each layer is a normal subgroup. Some years ago interesting characterizations of EL-groups were given by Ya. D. Polovickii ([12], Theorems 1 and 3) and we begin by giving another, more direct proof of his results. * In order to state these we need some further terminology. A group is said to be locally extremal and normal if every finite subset lies in a normal extremal subgroup. A direct product of groups is called prime-thin if for each prime p only a finite number of the direct factors contain elements of order p. PolovickiI’s theorem is as follows.