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A new approach to the k(GV)-problem

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 University601 University DriveSan Marcos, TX 78666USA e-mail: [email protected]
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Abstract

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This paper is concerned with the well-known and long-standing k(G V)-problem: If the finite group G acts faithfully and irreducibly on the finite GF(p)-module V and p does not divide the order of G, is the number k(GV) of conjugacy classes of the semidirect product GV bounded above by the order of V? Over the past two decades, through the work of numerous people, by using deep character theoretic arguments this question has been answered in the affirmative except for ρ = 5 for which it is still open. In this paper we suggest a new approach to the k(G V)-problem which is independent of most of the previous work on the problem and which is mainly group theoretical. To demonstrate the potential of the new line of attack we use it to solve the k(G V)-problem for solvable G and large ρ.

MSC classification

Secondary: 20C15: Ordinary representations and characters 20C20: Modular representations and characters
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 2 , October 2003 , pp. 193 - 220
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Dixon, J. D., ‘The fitting subgroup of a linear solvable group’, J. Austral. Math. Soc. 7 (1967), 417–424.CrossRefGoogle Scholar
[2]Gallagher, P. X., ‘The number of conjugacy classes in a finite group’, Math Z. 118 (1970), 175–179.CrossRefGoogle Scholar
[3]Gluck, D., ‘On the k(G V)-problem’, J. Algebra 89 (1984), 46–55.CrossRefGoogle Scholar
[4]Gluck, D. and Magaard, K., ‘The extraspecial case of the k(GV)-problem’, Trans. Amer. Math. Soc. 354 (2002), 287–333.Google Scholar
[5]Gluck, D. and Magaard, K., ‘The k(GV)-conjecture for modules in characteristic 31’, J. Algebra 250 (2002), 252–270.Google Scholar
[6]Goodwin, D., ‘Regular orbits of linear groups with an application to the k(GV)-problem I’, J. Algebra 227 (2000), 395–432.CrossRefGoogle Scholar
[7]Goodwin, D., ‘Regular orbits of linear groups with an application to the k(GV)-problem II’, J. Algebra 227 (2000), 433–473.CrossRefGoogle Scholar
[8]Guralnick, R. and Magaard, K., ‘On the minimal degree of a primitive permutation group’, J. Algebra 207 (1998), 127–145.CrossRefGoogle Scholar
[9]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[10]Huppert, B., Character theory of finite groups (de Gruyter, Berlin, 1998).Google Scholar
[11]Isaacs, I. M., Character theory of finite groups (Academic Press, New York, 1976).Google Scholar
[12]Keller, T. M., ‘Orbit sizes and character degrees, II’, J. Reine Angew. Math. 516 (1999), 27–114.CrossRefGoogle Scholar
[13]Knörr, R., ‘On the number of characters in a p-block of a p-solvable group’, Illinois J. Math. 28 (1984), 181–210.Google Scholar
[14]Koehler, C. and Pahlings, H., ‘Regular orbits and the k(gv)-problem’, in: Groups and computation, III, Ohio State Univ. Math. Res. Inst. Publ. 8 (de Gruyter, Berlin, 2001).Google Scholar
[15]Kovács, L. G. and Robinson, G. R., ‘On the number of conjugacy classes of a finite group’, J. Algebra 160 (1993), 441–460.Google Scholar
[16]Liebeck, M. W. and Pyber, L., ‘Upper bounds for the number of conjugacy classes of a finite group’, J. Algebra 198 (1997), 538–562.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]Murai, M., ‘A note on the number of irreducible characters in a p-block with normal defect group’, Proc. Japan Acad. Ser. A 59 (1983), 488–489.Google Scholar
[19]Murai, M., ‘A note on the number of irreducible characters in a p-block of a finite group’, Osaka J. Math. 21 (1984), 387–398.Google Scholar
[20]Nagao, H., ‘On a conjecture of Brauer for p-solvable groups’, J. Math. Osaka City Univ. 13 (1962), 35–38.Google Scholar
[21]Riese, U., ‘The quasisimple case of the k(gv)-conjecture’, J. Algebra 235 (2001), 45–65.CrossRefGoogle Scholar
[22]Riese, U. and Schmid, P., ‘Self-dual modules and real vectors for solvable groups’, J. Algebra 227 (2000), 159–171.Google Scholar
[23]Riese, U. and Schmid, P., ‘Real vectors for linear groups and the k(GV)-problem’, Preprint, 2001.Google Scholar
[24]Robinson, G. R., ‘Further reductions for the k(GV)-problem’, J. Algebra 195 (1997), 141–150.CrossRefGoogle Scholar
[25]Robinson, G. R. and Thompson, J. G., ‘On Brauer's k(B)-problem’, J. Algebra 184 (1996), 1143–1160.Google Scholar