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

Abstract

Suppose that $G$ is a finite group and $X={\mathit{\text{t}}}^G$   be a conjugacy class of an elements of order 3, $\mathit{\text{t}}\in G$.The A4-graph, is a simple undirected graph stand for $\mathit{\text{A}}{}_4(G,X)$, whose vertex set X and two vertices $\mathit{\text{x, y}}\in X$ are adjacent if they are different and satisfy  ${\mathit{\text{xy}}}^{-1}={\mathit{\text{yx}}}^{-1}$. In this article, the orbits under the action of $C_G(\mathit{\text{t}})$ on X are analyzed, along with  the description of the algebraic structure of the subgroup $<\mathit{\text{t,x}}>$  such that $\mathit{\text{x}}$ is a $C_G(\mathit{\text{t}})$ -orbit representative is provided.

Keywords

• Finite simple groups
• A4-graph
• connectivity
• cliques