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A new characterization of \(A_p\) where \(p\) and \(p-2\) are primes. (English) Zbl 1013.20010
Let \(G\) be a finite group and \(\pi_i\) (\(1\leq i\leq t\)) are all prime graph components of \(G\). Then \(|G|=m_1\cdots m_t\) for some coprime positive integers \(m_1,\dots,m_t\) such that \(\pi(m_i)=\pi_i\) (\(1\leq i\leq t\)). The integers \(m_1,\dots,m_t\) are called the order components of \(G\). The set \(\{m_1,\dots,m_t\}\) is denoted by \(OC(G)\). The authors prove the following theorem: if \(G\) is a finite group, and \(M=A_p\) where \(p\) and \(p-2\) are primes and \(OC(G)=OC(M)\), then \(G\cong M\). Earlier such a theorem was proved for the following groups \(M\): the groups \(^2B_2(q)\), \(^2G_2(q)\), \(^2F_4(q)\) (G.-Y. Chen [Sci. China, Ser. A 40, No. 8, 807-812 (1997; Zbl 0890.20015)]), the sporadic simple groups (G.-Y. Chen [Algebra Colloq. 3, No. 1, 49-58 (1996; Zbl 0845.20011)]), \(G_2(3^n)\) (G.-Y. Chen [J. Southwest China Norm. Univ. 21, No. 1, 47-51 (1996)]), \(E_8(q)\) (G.-Y. Chen [J. Southwest China Norm. Univ. 21, No. 3, 215-217 (1996)]), \(\text{PSL}(2,q)\) (G.-Y. Chen [Southeast Asian Bull. Math. 22, No. 3, 257-263 (1998; Zbl 0936.20010)]), \(F_4(2^n)\) (A. Iranmanesh and B. Khosravi [Far East J. Math. Sci. 2, No. 6, 853-859 (2000; Zbl 0972.20010)]). The proof of the theorem uses the classification of the prime graph components of the (known) finite simple groups (see J. S. Williams [J. Algebra 69, 487-513 (1981; Zbl 0471.20013)] and the reviewer [Mat. Sb. 180, No. 6, 787-797 (1989; Zbl 0691.20013)]). The theorem implies that, for the groups under consideration, Thompson’s well-known conjecture and Shi-Bi’s conjecture (see the problems 12.37 and 12.39 in [“Kourovka Notebook”, 12-th ed., Novosibirsk (1992; Zbl 0831.20003)], respectively) are true.

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.)