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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
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.
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