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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 $$M\cong G_2(q)$$.

MSC:
 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.)
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References:
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