Some results on the join graph of finite groups

Abstract

‎Let $G$ be a finite group which is not cyclic of prime power order‎. ‎The join graph $\Delta(G)$ of $G$ is a graph whose vertex set is the set of all proper subgroups of $G$‎, ‎which are not contained in the Frattini subgroup $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $G=\langle H‎, ‎K\rangle$‎. ‎Among other results‎, ‎we show that if $G$ is a finite cyclic group and $H$ is a finite group such that $\Delta(G)\cong\Delta(H)$‎, ‎then $H$ is cyclic‎. ‎Also we prove that $\Delta(G)\cong\Delta(A_5)$ if and only if $G\cong A_5$‎.

Keywords

• ‎finite group‎
• ‎join graph‎
• ‎cyclic group‎
• ‎alternating group