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A quantitative characterization of some finite simple groups through order and degree pattern. (English) Zbl 1316.20024

Summary: 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 \(\deg_G(p_i)\) of a vertex \(p_i\) is the number of edges incident on \(p_i\), and the \(h\)-tuple \((\deg_G(p_1),\deg_G(p_2),\ldots,\deg_G(p_h))\) 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.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D05 Finite simple groups and their classification
20D06 Simple groups: alternating groups and groups of Lie type
20D08 Simple groups: sporadic groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20G40 Linear algebraic groups over finite fields
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