Description of the A4-graph for elements of order three in certain miscellaneous groups

Abid, Wissam Fadhel; Msheree, Zainab Hasan; Abed, Mohammed Mukheef

doi:10.22075/ijnaa.2022.6963

Description of the A4-graph for elements of order three in certain miscellaneous groups

Document Type : Research Paper

Authors

Wissam Fadhel Abid ¹

Zainab Hasan Msheree ²

Mohammed Mukheef Abed ²

¹ Middle Technical University, Institute of Technology, Baghdad, Iraq

² Middle Technical University, Technical Instructors Training Institute, Iraq

10.22075/ijnaa.2022.6963

Abstract

Suppose that $G$ is a finite group and $X=\textit{t}^{G}$ be a conjugacy class of an elements of order 3, $ \textit{t}\in G $.The A4-graph, is a simple undirected graph stand for $\textit{A}{_{4}}(G,X)$, whose vertex set X and two vertices $ \textit{x, y}\in X $ are adjacent if they are different and satisfy $\textit{xy}^{-1} = \textit{yx}^{-1}$. In this article, the orbits under the action of $C_{G}(\textit{t})$ on X are analyzed, along with the description of the algebraic structure of the subgroup $<\textit{t,x}>$ such that $ \textit{x} $ is a $C_{G}(\textit{t})$ -orbit representative is provided.

Keywords

Finite simple groups

A4-graph

connectivity

cliques