On infinite groups with given properties of the norm of infinite subgroups

We investigate the relationship between the norm NG (∞) of infinite subgroups of an infinite group G and the structure of this group. We prove that NG (∞) is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if NG(∞) is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup NG(∞) has finite index in G. © 2001 Plenum Publishing Corporation.