Some results on the join graph of finite groups

Document Type : Research Paper


Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box 84156-83111, Isfahan, Iran


‎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 $G\cong A_5$‎.


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Volume 10, Issue 4 - Serial Number 4
December 2021
Pages 175-186
  • Receive Date: 31 May 2020
  • Revise Date: 04 October 2020
  • Accept Date: 05 October 2020
  • Published Online: 01 December 2021