Automorphisms of metacyclic groups

Abstract

A metacyclic group $H$ can be presented as $⟨\alpha ,\beta \mid {\alpha }^n=1,{\beta }^m={\alpha }^t,\beta \alpha {\beta }^{-1}={\alpha }^r⟩$ 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.

12 pages, accepted by Czech. Math. J

Cited by in corpus (2)

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