Crossed Product of Cyclic Groups

TL;DR

This paper explicitly describes all crossed products of two cyclic groups and provides a criterion for when an extension of a group by a cyclic group results in a cyclic group.

Contribution

It offers a complete classification of crossed products of cyclic groups and characterizes when extensions produce cyclic groups.

Findings

Explicit descriptions of all crossed products of two cyclic groups

Necessary and sufficient condition for cyclic group extensions

Enhanced understanding of group extension structures

Abstract

All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given.