Czechoslovak Mathematical Journal, first online, pp. 1-13


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
Classification MSC:  20D45
DOI:  10.21136/CMJ.2017.0656-16

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Affiliations:   Haimiao Chen, Beijing Technology and Business University, Fucheng Road 11/33, Beijing 10048, Haidian, China, e-mail: chenhm@pku.edu.cn; Yueshan Xiong, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan 430074, Hogshan, Hubei, China, e-mail: xiongyueshan@gmail.com; Zhongjian Zhu, Wenzhou University, 276 Xueyuan Middle Rd, Lucheng, Wenzhou 325035, Zhejiang, China, e-mail: zhuzhongjianzzj@126.com


 
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