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Keywords:
automorphism; metacyclic group; linear congruence equation
Summary:
A metacyclic group $H$ can be presented as $\langle \alpha ,\beta \colon \alpha ^{n}=1$, $\beta ^{m}=\alpha ^{t}$, $\beta \alpha \beta ^{-1}=\nobreak \alpha ^{r}\rangle$ 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.
References:
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