The classification of finite groups by using iteration digraphs

Czechoslovak Mathematical Journal

Uzma ¹ 

[1] Muqadas Moeen, University of the Punjab, New Campus, Lahore, 54000, Punjab, Pakistan

关键词: $2$-group;    generalized quaternion group;    iteration digraph;    cycle;    indegree;    fixed point;    regular digraph;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

A digraph is associated with a finite group by utilizing the power map ${fG \rightarrow G}$ 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.

【 授权许可】

Unknown