A New Characterization of Sporadic Groups

TL;DR

This paper introduces a new characterization of sporadic simple groups based on the disconnection of their prime graph, unifying previous criteria and providing a clear group identification method.

Contribution

It establishes that a finite group is isomorphic to a sporadic simple group if and only if it has the same order and a disconnected prime graph, unifying multiple existing characterizations.

Findings

Characterization of sporadic groups via prime graph disconnection

Unification of previous sporadic group characterizations

Provides a new criterion for identifying sporadic simple groups

Abstract

Let $G$ be a finite group, $n$ a positive integer. $\pi(n)$ denotes the set of all prime divisors of $n$ and $\pi(G)=\pi(|G|)$. The prime graph $\Gamma(G)$ of $G$, defined by Grenberg and Kegel, is a graph whose vertex set is $\pi(G)$, two vertices $p,\ q$ in $\pi(G)$ 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 $\Gamma(G)$ 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.