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Further reflections on Thompson’s conjecture. (English) Zbl 0931.20020
Let \(G\) be a finite group and let \(N(G)\) be the set of orders of all conjugacy classes of \(G\). J. G. Thompson has conjectured that if \(Z(G)=1\) and \(M\) is a non-abelian simple group with \(N(G)=N(M)\) then \(G\cong M\). In previous papers [Algebra Colloq. 3, No. 1, 49-58 (1996; Zbl 0845.20011) and J. Algebra 185, No. 1, 184-193 (1996; Zbl 0861.20018)], the author proved Thompson’s conjecture in the case when \(M\) is a sporadic simple group or a simple group with at least three prime graph components. In this paper, he proves Thompson’s conjecture for \(M\cong G_2(q)\).

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D06 Simple groups: alternating groups and groups of Lie type
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Full Text: DOI
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