# Groups in which each subgroup is commensurable with a normal subgroup

## TL;DR

This paper studies a special class of groups called cn-groups, showing that under certain conditions they are finite-by-abelian-by-finite, and provides detailed descriptions using automorphisms of abelian groups.

## Contribution

It characterizes cn-groups with periodic images as finite-by-abelian-by-finite and analyzes their structure via automorphisms of abelian groups.

## Key findings

- Cn-groups with locally finite periodic images are finite-by-abelian-by-finite.
- Such groups with bounded index are also finite-by-abelian-by-finite.
- Automorphisms of abelian groups help describe the structure of these groups.

## Abstract

A group G is a cn-group if for each subgroup H of G there exists a normal subgroup N of G such that the index of both H and N in HN is finite. The class of cn-groups contains properly the classes of core- finite groups and that of groups in which each subgroup has finite index in a normal subgroup. In the present paper it is shown that a cn-group whose periodic images are locally finite is finite-by-abelian-by-finite. Such groups are then described into some details by considering automorphisms of abelian groups. Finally, it is shown that if G is a locally graded group with the property that the above index is bounded independently of H, then G is finite-by-abelian-by-finite.