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OD-characterization of all simple groups whose orders are less than \(10^8\). (English) Zbl 1165.20010

Let \(G\) be a finite group, \(\pi(G)\) be the set of prime divisors of \(|G|\) and \(\pi_i\) (\(1\leq i\leq s\)) (\(s=s(G)\)) be all prime graph components of \(G\). Then \(|G|=m_1\cdots m_s\) for some coprime positive integers \(m_1,\dots,m_s\) such that \(\pi(m_i)=\pi_i\) (\(1\leq i\leq s\)). The integers \(m_1,\dots,m_s\) are called the order components of \(G\). The set \(\{m_1,\dots,m_s\}\) is denoted by \(\text{OC}(G)\). If \(p\in\pi(G)\) then \(\deg(p)\) denotes the degree of the vertex \(p\) in the prime graph of \(G\). According to A. R. Moghaddamfar, A. R. Zokayi and M. R. Darafsheh [Algebra Colloq. 12, No. 3, 431-442 (2005; Zbl 1072.20015)], it is defined the set \(\text{Deg}(G)=\{\deg(p_1),\dots,\deg(p_k)\}\), where \(\pi(G)=\{p_1,p_2,\dots,p_k\}\) and \(p_1<p_2<\cdots<p_k\), called the degree pattern of \(G\).
In the present paper, the authors prove that if \(M\) is a simple group whose order is less than \(10^8\) and \(G\) is a finite group with the same order and degree pattern as \(M\) then either \(G\cong M\), or \(M\in\{A_{10},U_4(2)\}\).

MSC:

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

Citations:

Zbl 1072.20015
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References:

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