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Groups with few conjugacy classes

Part of: Representation theory of groups

Published online by Cambridge University Press:  31 March 2011

László Héthelyi ,
Erzsébet Horváth ,
Thomas Michael Keller  and
Attila Maróti
László Héthelyi
Affiliation:
Department of Algebra, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3–9, 1521 Budapest, Hungary (hethelyi@math.bme.hu; he@math.bme.hu)
Erzsébet Horváth
Affiliation:
Department of Algebra, Institute of Mathematics, Budapest University of Technology and Economics, Műegyetem rkp. 3–9, 1521 Budapest, Hungary (hethelyi@math.bme.hu; he@math.bme.hu)
Thomas Michael Keller
Affiliation:
Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666, USA (keller@txstate.edu)
Attila Maróti
Affiliation:
MTA Alfréd Rényi Institute of Mathematics, Réaltanoda utca 13–15, 1053 Budapest, Hungary (maroti@renyi.hu)
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Abstract

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Let G be a finite group, let p be a prime divisor of the order of G and let k(G) be the number of conjugacy classes of G. By disregarding at most finitely many non-solvable p-solvable groups G, we have with equality if and only if if is an integer, and CG(Cp) = Cp. This extends earlier work of Héthelyi, Külshammer, Malle and Keller.

Keywords

finite groupconjugacy classeslower boundFrobenius group

MSC classification

Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 423 - 430
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Carter, R. W., Finite groups of Lie type (Wiley, 1985).Google Scholar
2.Conway, J. H., Curtis, R. T, Norton, S. P, Parker, R. A and Wilson, R. A., Atlas of finite groups: maximal subgroups and ordinary characters for simple groups (Oxford University Press, 1985).Google Scholar
3.Dornhoff, L., Group representation theory (North-Holland, Amsterdam, 1982).Google Scholar
4.GAP Group, GAP—groups, algorithms, and programming, Version 4.4 (available at www.gap-system.org; 2005).Google Scholar
5.Héthelyi, L. and Külshammer, B., On the number of conjugacy classes of a finite solvable group, Bull. Lond. Math. Soc. 32 (2000), 668–672.CrossRefGoogle Scholar
6.Keller, T. M., Lower bounds for the number of conjugacy classes of finite groups, Math. Proc. Camb. Phil. Soc. 147 (2009), 567–577.CrossRefGoogle Scholar
7.Kleidman, P. B. and Liebeck, M. W., The subgroup structure of the finite classical groups, London Mathematical Society Lecture Notes, Volume 129 (Cambridge University Press, 1990).Google Scholar
8.Landau, E., Über die Klassenzahl der binären quadratischen Formen von negativer Diskriminante, Math. Annalen 56 (1903), 671–676.CrossRefGoogle Scholar
9.Malle, G., Fast-einfache Gruppen mit langen Bahnen in absolut irreduzibler Operation, J. Alg. 300 (2006), 655–672.CrossRefGoogle Scholar
10.Manz, O. and Wolf, T., Representations of solvable groups (Cambridge University Press, 1993).CrossRefGoogle Scholar
11.Pintz, J., Landau's problems on primes, J. Théorie Nombres Bordeaux 21(2) (2009), 357–404.CrossRefGoogle Scholar
12.Pyber, L., Finite groups have many conjugacy classes, J. Lond. Math. Soc. 46 (1992), 239–249.CrossRefGoogle Scholar
13.Seager, S. M., The rank of a finite primitive solvable permutation group, J. Alg. 105 (1987), 389–394.CrossRefGoogle Scholar
14.Vera López, A. and Vera López, J., Classification of finite groups according to the number of conjugacy classes, I, Israel J. Math. 51 (1985), 305–338.CrossRefGoogle Scholar
15.Vera López, A. and Vera López, J., Classification of finite groups according to the number of conjugacy classes, II, Israel J. Math. 56 (1986), 188–221.CrossRefGoogle Scholar