Some results on the main supergraph of finite groups

Abstract

Let \(G\) be a finite group. The main supergraph \(\mathcal{S}(G)\) is a graph with vertex set \(G\) in which two vertices \(x\) and \(y\) are adjacent if and only if \(o(x) \mid o(y)\) or \(o(y)\mid o(x)\). In this paper, we will show that \(G\cong \mathrm{PSL}(2,p)\) or \(\mathrm{PGL}(2,p)\) if and only if \(\mathcal{S}(G)\cong \mathcal{S}(\mathrm{PSL}(2,p))\) or \(\mathcal{S}(\mathrm{PGL}(2,p))\), respectively. Also, we will show that if \(M\) is a sporadic simple group, then \(G\cong M\) if only if \(\mathcal{S}(G)\cong \mathcal{S}(M)\).