Czechoslovak Mathematical Journal, Vol. 68, No. 3, pp. 803-815, 2018
Automorphisms of metacyclic groups
Haimiao Chen, Yueshan Xiong, Zhongjian Zhu
Received December 31, 2016. First published December 7, 2017.
Abstract: A metacyclic group $H$ can be presented as $\langle\alpha,\beta \alpha^n=1$, $ \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.
Keywords: automorphism; metacyclic group; linear congruence equation