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

doi:10.1285/i15900932v34n2p91

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

A. R. Moghaddamfar, S. Rahbariyan

Abstract

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

Abstract