The classification of finite groups by using iteration digraphs

Ahmad, Uzma, and Moeen, Muqadas. "The classification of finite groups by using iteration digraphs." Czechoslovak Mathematical Journal 66.4 (2016): 1103-1117. <http://eudml.org/doc/287529>.

@article{Ahmad2016,
abstract = {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.},
author = {Ahmad, Uzma, Moeen, Muqadas},
journal = {Czechoslovak Mathematical Journal},
keywords = {$2$-group; generalized quaternion group; iteration digraph; cycle; indegree; fixed point; regular digraph; 2-group; generalized quaternion group; iteration digraph; cycle; indegree; fixed point; regular digraph},
language = {eng},
number = {4},
pages = {1103-1117},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The classification of finite groups by using iteration digraphs},
url = {http://eudml.org/doc/287529},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Ahmad, Uzma
AU - Moeen, Muqadas
TI - The classification of finite groups by using iteration digraphs
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 4
SP - 1103
EP - 1117
AB - 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.
LA - eng
KW - $2$-group; generalized quaternion group; iteration digraph; cycle; indegree; fixed point; regular digraph; 2-group; generalized quaternion group; iteration digraph; cycle; indegree; fixed point; regular digraph
UR - http://eudml.org/doc/287529
ER -