Automorphisms of metacyclic groups

TL;DR

This paper characterizes automorphisms of metacyclic groups by providing necessary and sufficient conditions on the parameters defining endomorphisms, enhancing understanding of their algebraic structure.

Contribution

It offers a complete characterization of automorphisms of metacyclic groups through explicit conditions on endomorphism parameters.

Findings

Derived necessary and sufficient conditions for endomorphisms to be automorphisms.

Provided a framework to identify automorphisms based on group presentation parameters.

Enhanced understanding of the automorphism structure of metacyclic groups.

Abstract

A metacyclic group $H$ can be presented as $\langle \alpha,\beta\mid \alpha^{n}=1, \ \beta^{m}=\alpha^{t}, \ \beta\alpha\beta^{-1}=\alpha^{r}\rangle$ for some $n,m,t,r$. Each endomorphism $\sigma$ of $H$ is determined by $\sigma(\alpha)=\alpha^{x_{1}}\beta^{y_{1}}, \sigma(\beta)=\alpha^{x_{2}}\beta^{y_{2}}$ for some integers $x_{1},x_{2},y_{1},y_{2}$. We give sufficient and necessary conditions on $x_{1},x_{2},y_{1},y_{2}$ for $\sigma$ to be an automorphism.