Automorphisms of metacyclic groups

Haimiao Chen; Yueshan Xiong; Zhongjian Zhu

Czechoslovak Mathematical Journal (2018)

  • Volume: 68, Issue: 3, page 803-815
  • ISSN: 0011-4642

Abstract

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A metacyclic group H can be presented as α , β : α n = 1 , β m = α t , β α β - 1 = α r for some n , m , t , r . Each endomorphism σ of H is determined by σ ( α ) = α x 1 β y 1 , σ ( β ) = α x 2 β 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 σ to be an automorphism.

How to cite

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Chen, Haimiao, Xiong, Yueshan, and Zhu, Zhongjian. "Automorphisms of metacyclic groups." Czechoslovak Mathematical Journal 68.3 (2018): 803-815. <http://eudml.org/doc/294178>.

@article{Chen2018,
abstract = {A metacyclic group $H$ can be presented as $\langle \alpha ,\beta \colon \alpha ^\{n\}=1$, $ \beta ^\{m\}=\alpha ^\{t\}$, $\beta \alpha \beta ^\{-1\}=\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.},
author = {Chen, Haimiao, Xiong, Yueshan, Zhu, Zhongjian},
journal = {Czechoslovak Mathematical Journal},
keywords = {automorphism; metacyclic group; linear congruence equation},
language = {eng},
number = {3},
pages = {803-815},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Automorphisms of metacyclic groups},
url = {http://eudml.org/doc/294178},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Chen, Haimiao
AU - Xiong, Yueshan
AU - Zhu, Zhongjian
TI - Automorphisms of metacyclic groups
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 3
SP - 803
EP - 815
AB - A metacyclic group $H$ can be presented as $\langle \alpha ,\beta \colon \alpha ^{n}=1$, $ \beta ^{m}=\alpha ^{t}$, $\beta \alpha \beta ^{-1}=\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.
LA - eng
KW - automorphism; metacyclic group; linear congruence equation
UR - http://eudml.org/doc/294178
ER -

References

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  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. (2017) 
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  6. Golasiński, M., Gonçalves, D. L., 10.1007/s00229-008-0233-4, Manuscripta Math. 128 (2009), 251-273. (2009) Zbl1160.20017MR2471317DOI10.1007/s00229-008-0233-4
  7. Hempel, C. E., 10.1080/00927870008827063, Commun. Algebra 28 (2000), 3865-3897. (2000) Zbl0993.20013MR1767595DOI10.1080/00927870008827063
  8. Zassenhaus, H. J., The Theory of Groups, Chelsea Publishing Company, New York (1958). (1958) Zbl0083.24517MR0091275

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