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On Thompson’s conjecture. (English) Zbl 0861.20018
For a finite group \(G\), \(N(G)\) denotes the set of the orders of conjugacy classes of \(G\). J. G. Thompson has conjectured that if \(G\) is a finite group with \(Z(G)=1\) and \(M\) is a finite nonabelian simple group with \(N(G)=N(M)\), then \(G\cong M\). In [Proc. China Assoc. Sci. Tech. First. Acad. Ann. Meeting of Youths, 1-6, Chin. Sci. Tech. Press, Beijing (1992)], the author has proved that if \(M\) is a sporadic simple group, then Thompson’s conjecture is valid. In this paper, he proves the validity of the conjecture for each finite simple group \(M\) having at least three prime graph components. The proof is based on the classification of prime graph components of (known) finite simple groups given by 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)].

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.)
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