Some Simple Groups Which are Determined by Their Orders and Degree-Patterns

Abstract

Let [Formula: see text] be a finite group and [Formula: see text] the set of all irreducible complex characters of [Formula: see text]. Let [Formula: see text] be the set of all irreducible complex character degrees of [Formula: see text] and denote by [Formula: see text] the set of all primes which divide some character degree of [Formula: see text]. The character-prime graph [Formula: see text] associated to [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] and there is an edge between two distinct primes [Formula: see text] and [Formula: see text] if and only if the product [Formula: see text] divides some character degree of [Formula: see text]. We show that the finite nonabelian simple groups [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are uniquely determined by their degree-patterns and orders.