| Title:
|
The classification of finite groups by using iteration digraphs (English) |
| Author:
|
Ahmad, Uzma |
| Author:
|
Moeen, Muqadas |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
1103-1117 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A digraph is associated with a finite group by utilizing the power map ${f\colon G \rightarrow G}$ defined by $f(x)=x^{k}$ for all $x\in G$, where $k$ is a fixed natural number. It is denoted by $\gamma _{G}(n,k)$. In this paper, the generalized quaternion and $2$-groups are studied. The height structure is discussed for the generalized quaternion. The necessary and sufficient conditions on a power digraph of a $2$-group are determined for a $2$-group to be a generalized quaternion group. Further, the classification of two generated $2$-groups as abelian or non-abelian in terms of semi-regularity of the power digraphs is completed. (English) |
| Keyword:
|
$2$-group |
| Keyword:
|
generalized quaternion group |
| Keyword:
|
iteration digraph |
| Keyword:
|
cycle |
| Keyword:
|
indegree |
| Keyword:
|
fixed point |
| Keyword:
|
regular digraph |
| MSC:
|
05C20 |
| MSC:
|
05C25 |
| MSC:
|
05C50 |
| MSC:
|
20B25 |
| MSC:
|
20D15 |
| idZBL:
|
Zbl 06674864 |
| idMR:
|
MR3572925 |
| DOI:
|
10.1007/s10587-016-0312-8 |
| . |
| Date available:
|
2016-11-26T20:52:37Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145921 |
| . |
| Reference:
|
[1] Ahmad, U., Husnine, S.: The power digraphs of finite groups.(to appear) in Util. Math. |
| Reference:
|
[2] Ahmad, U., Husnine, S.: Characterization of power digraphs modulo $n$.Commentat. Math. Univ. Carol. 52 (2011), 359-367. Zbl 1249.11002, MR 2843229 |
| Reference:
|
[3] Ahmad, U., Moeen, M.: The digraphs arising by the power maps of generalized quaternion groups.(to appear) in J. Algebra Appl. DOI:10.1142/S0219498817501791. 10.1142/S0219498817501791 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[7] Husnine, S. M., Ahmad, U., Somer, L.: On symmetries of power digraphs.Util. Math. 85 (2011), 257-271. Zbl 1251.05066, MR 2840802 |
| Reference:
|
[8] Janko, Z.: A classification of finite $2$-groups with exactly three involutions.J. Algebra 291 (2005), 505-533. Zbl 1081.20025, MR 2163481, 10.1016/j.jalgebra.2005.02.007 |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
