A New Characterization of Sporadic Groups

2020/09/16 by Wang, Zhongbi, Lv, Heng, Yan, Yanxiong +1
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2009.07490

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≅ 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.

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