A Quantitative Characterization of Some Finite Simple Groups Through Order and Degree Pattern

Abstract

Let be a finite group with , where are prime numbers and are natural numbers. The prime graph of is a simple graph whose vertex set is and two distinct primes and are joined by an edge if and only if has an element of order . The degree of a vertex is the number of edges incident on , and the -tuple is called the degree pattern of . We say that the problem of OD-characterization is solved for a finite group if we determine the number of pairwise non-isomorphic finite groups with the same order and degree pattern as . The purpose of this paper is twofold. First, it completely solves the OD-characterization problem for every finite non-Abelian simple groups their orders having prime divisors at most 17. Second, it provides a list of finite (simple) groups for which the problem of OD-characterization have been already solved.