A New Characterization of Sporadic Groups

Abstract

Let $G$ be a finite group, $n$ a positive integer. $π(n)$ denotes the set of all prime divisors of $n$ and $π(G)=π(|G|)$. The prime graph $Γ(G)$ of $G$, defined by Grenberg and Kegel, is a graph whose vertex set is $π(G)$, two vertices $p,\ q$ in $π(G)$ joined by an edge if and only if $G$ contains an element of order $pq$. In this article, a new characterization of sporadic simple groups is obtained, that is, if $G$ is a finite group and $S$ a sporadic simple group. Then $G\cong S$ if and only if $|G|=|S|$ and $Γ(G)$ is disconnected. This characterization unifies the several characterizations that can conclude the group has non-connected prime graphs, hence several known characterizations of sporadic simple groups become the corollaries of this new characterization.