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Lower bounds for the number of conjugacy classes of finite groups

Published online by Cambridge University Press:  15 June 2009

THOMAS MICHAEL KELLER
THOMAS MICHAEL KELLER*
Affiliation:
Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666, U.S.A. e-mail: [email protected]
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Abstract

In 2000, L. Héthelyi and B. Külshammer proved that if p is a prime number dividing the order of a finite solvable group G, then G has at least conjugacy classes. In this paper we show that if p is large, the result remains true for arbitrary finite groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 3 , November 2009 , pp. 567 - 577
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Aschbacher, M.Finite Group Theory, (Cambridge University Press, 1986).Google Scholar
[2]Babai, L. and Pyber, L.Permutation groups without exponentially many orbits on the power set. J. Combin. Theory Ser. A 66 (1994), 160–168.CrossRefGoogle Scholar
[3]Bertram, E. A.Lower bounds for the number of conjugacy classes in finite groups. Ischia Group Theory (2004), 95–117. Contemp. Math. 402 (AMS, 2006).CrossRefGoogle Scholar
[4]Dixon, J. D.The Structure of Linear Groups. (Van Nostrand-Reinhold, 1971).Google Scholar
[5]Gambini-Weigel, A. and Weigel, T. S.On the orders of primitive linear p′-groups. Bull. Austral. Math. Soc. 48 (1993), 495–521.CrossRefGoogle Scholar
[6]Gluck, D. and Magaard, K.Base sizes and regular orbits for coprime affine permutation groups. J. London Math. Soc. (2) 58 (1998), 603–618.CrossRefGoogle Scholar
[7]Héthelyi, L. and Külshammer, B.On the number of conjugacy classes of a finite solvable group. Bull. London Math. Soc. 32 (2000), 668–672.CrossRefGoogle Scholar
[8]Huppert, B.Character Theory of Finite Groups (deGruyter, Berlin, 1998).CrossRefGoogle Scholar
[9]Huppert, B.Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[10]Huppert, B.Zweifach transitive auflösbare permutationsgruppen. Math. Z. 68 (1957), 126–150.CrossRefGoogle Scholar
[11]Keller, T. M.The k(GV)-problem revisited: J. Austral. Math. Soc. 79 (2005), 257–276.CrossRefGoogle Scholar
[12]Manz, O. and Wolf, T. R.Representations of solvable groups. London Math. Soc. Lecture Notes Series 185, (Cambridge University Press, 1993).CrossRefGoogle Scholar
[13]Maróti, A.On elementary lower bounds for the partition function. Integers 3 (2003), #A 10 (9 pages).Google Scholar
[14]Malle, G.Fast-einfache Gruppen mit langen Bahnen in absolut irreduzibler operation. J. Algebra 300 (2006), 655–672.CrossRefGoogle Scholar
[15]Praeger, C. E. and Saxl, J. On the orders of primitive permutation groups. Bull. London Math. Soc. (12) (1980), 303–307.CrossRefGoogle Scholar
[16]Pyber, L.Finite groups have many conjugacy classes. J. London Math. Soc. (2) 46 (1992), 239–249.CrossRefGoogle Scholar
[17]Robinson, G. R. and Thompson, J. G.On Brauer's k(B)-problem. J. Algebra 184 (1996), 1143–1160.CrossRefGoogle Scholar
[18]Weigel, T. S. personal email communication, July 2006.Google Scholar