Geodesic
Automorphisms of metacyclic groups
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 803-815 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/CMJ.2017.0656-16
Classification : 20D45
Keywords: automorphism; metacyclic group; linear congruence equation 
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     title = {Automorphisms of metacyclic groups},
     journal = {Czechoslovak Mathematical Journal},
     pages = {803--815},
     year = {2018},
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     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0656-16/}
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Chen, Haimiao; Xiong, Yueshan; Zhu, Zhongjian. Automorphisms of metacyclic groups. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 803-815. doi: 10.21136/CMJ.2017.0656-16

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