Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T02:45:46.152Z Has data issue: false hasContentIssue false
  • English
  • Français

The k(GV)-problem revisited

Part of: Representation theory of groups

Published online by Cambridge University Press:  09 April 2009

Thomas Michael Keller
Thomas Michael Keller
Affiliation:
Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666 USA, e-mail: keller@txstate.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose that the finite group G acts faithfully and irreducibly on the finite G-module V of characteristic p not dividing |G|. The well-known k(GV)-problem states that in this situation, if k(G V) is the number of conjugacy classes of the semidirect product GV, then k(GV) ≤ |V|. For p—solvable groups, this is equivalent to Brauer's famous k(B)-problem. In 1996, Robinson and Thompson proved the k(GV) problem for large p. This ultimately led to a complete proof of the k(GV)-problem. In this paper, we present a new proof of the k(G V)-problem for large p.

MSC classification

Secondary: 20C15: Ordinary representations and characters 20C20: Modular representations and characters
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 2 , October 2005 , pp. 257 - 276
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Aschbacher, M., ‘On the maximal subgroups of the finite classical groups’, Invent. Math. 76 (1984), 469–514.CrossRefGoogle Scholar
[2]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite simple groups (Clarendon Press, Oxford, 1985).Google Scholar
[3]Dornhoff, L., Group representation theory. Part A: Ordinary representation theory (Marcel Dekker, New York, 1971).Google Scholar
[4]Frohardt, D. and Magaard, K., ‘Composition factors of monodromy groups’, Ann. of Math. (2) 154 (2001), 327–345.CrossRefGoogle Scholar
[5]Gallagher, P. X., ‘The number of conjugacy classes in a finite group’, Math. Z. 118 (1970), 175–179.CrossRefGoogle Scholar
[6]Weigel, A. Gambini and Weigel, T. S., ‘On the orders of primitive linear p'-groups’, Bull. Austral. Math. Soc. 48 (1993), 495–521.CrossRefGoogle Scholar
[7]Gluck, D. and Magaard, K., ‘Base sizes and regular orbits for coprime affine permutation groups’, J. London Math. Soc. 58 (1998), 603–618.CrossRefGoogle Scholar
[8]Gluck, D., and Magaard, K., Riese, U. and Schmid, P., ‘The solution of the k(G V)-problem’, J. Algebra 279 (2004), 694–719.CrossRefGoogle Scholar
[9]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[10]Huppert, B., Character theory of finite groups (deGruyter, Berlin, 1998).CrossRefGoogle Scholar
[11]Keller, T. M., ‘A new approach to the k(G V)-problem’, J. Austral. Math. Soc. 75 (2003), 193–219.CrossRefGoogle Scholar
[12]Keller, T. M., ‘Orbits in finite group actions’, in: Groups St. Andrews 2001 in Oxford, London Math. Soc. Lecture Notes Series 305 (Cambridge University Press, 2003) pp. 306–331.CrossRefGoogle Scholar
[13]Liebeck, M. W., ‘Regular orbits of linear groups’, J. Algebra 184 (1996), 1136–1142.CrossRefGoogle Scholar
[14]Liebeck, M. W. and Pyber, L., ‘Upper bounds for the number of conjugacy classes of a finite group’, J. Algebra 198 (1997), 538–562.CrossRefGoogle Scholar
[15]Liebeck, M. W. and Shalev, A., ‘Diameters of finite simple groups: sharp bounds and applications’, Ann. of Math. (2) 154 (2001), 383–406.CrossRefGoogle Scholar
[16]Lyons, R., Finite group theory, Notes for Math. 555, unpublished, 1999.Google Scholar
[17]Manz, O. and Wolf, T. R., Representations of solvable groups, London Math. Soc. Lecture Notes Series 185 (Cambridge University Press, 1993).CrossRefGoogle Scholar
[18]Navarro, G., Characters and blocks of finite groups, London Math. Soc. Lecture Notes Series 250 (Cambridge University Press, 1998).CrossRefGoogle Scholar
[19]Pak, I., ‘On probability of generating a finite group’, unpublished manuscript, 1999.Google Scholar
[20]Pálfy, P. P. and Pyber, L., ‘Small groups of automorphisms’, Bull. London Math. Soc. 30 (1998), 386–390.CrossRefGoogle Scholar
[21]Riese, U., ‘On the number of irreducible characters in a block’, Algebra Colloq. 10 (2003), 381–390; see http://www-math.mit.edu/~pak/research.htmlGoogle Scholar
[22]Robinson, G. R. and Thompson, J. G., ‘On Brauer's k(B)-problem’, J. Algebra 184 (1996), 1143–1160.CrossRefGoogle Scholar
[23]Shalev, A., ‘Fixed point ratios, character ratios, and Cayley graphs’, in: Combinatorial and computational Algebra (Hong Kong, 1999), Contemp. Math. 264 (2000) pp. 49–59.Google Scholar
[24]Wagner, A., ‘The faithful linear representations of least degree of Sn and An over a field of odd characteristic’, Math. Z. 154 (1977), 103–114.CrossRefGoogle Scholar