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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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[1] Chen, G.Y., On Thompson’s conjecture—for sporadic groups (Chinese), Proceedings of the China association of science and technology, first Academic annual meeting of youths, (1992), Chinese Science and Technology Press Beijing, p. 1-6
[2] Chen, G.Y., On the structure of Frobenius group and 2-Frobenius group, J. southwest China normal univ., 20, 485-487, (1995)
[3] Chen, G.Y., A new characterization of sporadic simple groups, Algebra colloq., 3, 49-58, (1996) · Zbl 0845.20011
[4] Chen, G.Y., On Thompson’s conjecture, J. algebra, 185, 184-193, (1996) · Zbl 0861.20018
[5] Conway, J.H.; Curtis, R.T., Atlas of finite groups, (1985), Clarendon Oxford · Zbl 0568.20001
[6] Kondratev, A.S., Prime graph components of finite simple groups, Mat. USSR sb., 67, 235-247, (1990) · Zbl 0698.20009
[7] Shi, W.J.; Bi, J.X., A characteristic property of each finite projective special group, Lecture notes in mathematics, (1990), Springer-Verlag Berlin/New York, p. 171-180
[8] Williams, J.S., Prime graph components of finite groups, J. algebra, 69, 487-513, (1981) · Zbl 0471.20013
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