| Title:
|
Automorphisms of metacyclic groups (English) |
| Author:
|
Chen, Haimiao |
| Author:
|
Xiong, Yueshan |
| Author:
|
Zhu, Zhongjian |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
68 |
| Issue:
|
3 |
| Year:
|
2018 |
| Pages:
|
803-815 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
automorphism |
| Keyword:
|
metacyclic group |
| Keyword:
|
linear congruence equation |
| MSC:
|
20D45 |
| idZBL:
|
Zbl 06986973 |
| idMR:
|
MR3851892 |
| DOI:
|
10.21136/CMJ.2017.0656-16 |
| . |
| Date available:
|
2018-08-09T13:14:41Z |
| Last updated:
|
2020-10-05 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147369 |
| . |
| Reference:
|
[1] Bidwell, J. N. S., Curran, M. J.: The automorphism group of a split metacyclic $p$-group.Arch. Math. 87 (2006), 488-497. Zbl 1116.20016, MR 2283679, 10.1007/s00013-006-1899-z |
| Reference:
|
[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. |
| Reference:
|
[3] Curran, M. J.: The automorphism group of a split metacyclic 2-group.Arch. Math. 89 (2007), 10-23. Zbl 1125.20015, MR 2322775, 10.1007/s00013-007-2107-5 |
| Reference:
|
[4] Curran, M. J.: The automorphism group of a nonsplit metacyclic $p$-group.Arch. Math. 90 (2008), 483-489. Zbl 1149.20019, MR 2415289, 10.1007/s00013-008-2583-2 |
| Reference:
|
[5] Davitt, R. M.: The automorphism group of a finite metacyclic $p$-group.Proc. Am. Math. Soc. 25 (1970), 876-879. Zbl 0202.02501, MR 0285594, 10.2307/2036770 |
| Reference:
|
[6] Golasiński, M., Gonçalves, D. L.: On automorphisms of split metacyclic groups.Manuscripta Math. 128 (2009), 251-273. Zbl 1160.20017, MR 2471317, 10.1007/s00229-008-0233-4 |
| Reference:
|
[7] Hempel, C. E.: Metacyclic groups.Commun. Algebra 28 (2000), 3865-3897. Zbl 0993.20013, MR 1767595, 10.1080/00927870008827063 |
| Reference:
|
[8] Zassenhaus, H. J.: The Theory of Groups.Chelsea Publishing Company, New York (1958). Zbl 0083.24517, MR 0091275 |
| . |
