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


Let G be a finite group with |G|=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_h^{\alpha_h}, where p_1<p_2<\cdots<p_h are prime numbers and \alpha_1, \alpha_2, \ldots, \alpha_h, h are natural numbers. The prime graph \Gamma(G) of G is a simple graph whose vertex set is \{p_1, p_2, \ldots, p_h\} and two distinct primes p_i and p_j are joined by an edge if and only if G has an element of order p_ip_j. The degree {\rm deg}_G(p_i) of a vertex p_i is the number of edges incident on p_i, and the h-tuple ({\rm deg}_G(p_1), {\rm deg}_G(p_2), \ldots, {\rm deg}_G(p_h)) is called the degree pattern of G. We say that the problem of OD-characterization is solved for a finite group G if we determine the number of pairwise non-isomorphic finite groups with the same order and degree pattern as G. 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.

DOI Code: 10.1285/i15900932v34n2p91

Keywords: Prime graph; degree pattern; OD-characterization

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