Products of locally cyclic groups

TL;DR

This paper classifies groups formed by the product of two locally cyclic subgroups, detailing their structure in various periodicity cases and showing that products of certain permutable subgroups are locally supersoluble.

Contribution

It provides a complete classification of groups as products of two locally cyclic groups, extending previous work to include periodic and torsion-free cases.

Findings

Complete classification of such groups in different periodicity cases

Product of permutable periodic locally cyclic subgroups is locally supersoluble

Structural descriptions for groups with locally cyclic factors

Abstract

We consider groups of the form $G=AB$ with two locally cyclic subgroups $A$ and $B$. The structure of these groups is determined in the cases when $A$ and $B$ are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case when $A$ and $B$ are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic groups. It is also shown that the product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group.