On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

Abstract

We investigate the relationship between the norm N _G(∞) of infinite subgroups of an infinite group G and the structure of this group. We prove that N _G(∞) is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if N _G(∞) is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup N _G(∞) has finite index in G