Geodesic
Automorphisms of metacyclic groups
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 3, pp. 803-815
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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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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

[1] Bidwell, J. N. S., Curran, M. J.: The automorphism group of a split metacyclic $p$-group. Arch. Math. 87 (2006), 488-497. | DOI | MR | JFM

[2] Chen, H.-M.: Reduction and regular t-balanced Cayley maps on split metacyclic 2-groups. Available at ArXiv:1702.08351 [math.CO] (2017), 14 pages.

[3] Curran, M. J.: The automorphism group of a split metacyclic 2-group. Arch. Math. 89 (2007), 10-23. | DOI | MR | JFM

[4] Curran, M. J.: The automorphism group of a nonsplit metacyclic $p$-group. Arch. Math. 90 (2008), 483-489. | DOI | MR | JFM

[5] Davitt, R. M.: The automorphism group of a finite metacyclic $p$-group. Proc. Am. Math. Soc. 25 (1970), 876-879. | DOI | MR | JFM

[6] Golasiński, M., Gonçalves, D. L.: On automorphisms of split metacyclic groups. Manuscripta Math. 128 (2009), 251-273. | DOI | MR | JFM

[7] Hempel, C. E.: Metacyclic groups. Commun. Algebra 28 (2000), 3865-3897. | DOI | MR | JFM

[8] Zassenhaus, H. J.: The Theory of Groups. Chelsea Publishing Company, New York (1958). | MR | JFM

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