A new approach to thek(GV)-problem

Abstract

AbstractThis paper is concerned with the well-known and long-standingk(G V)-problem: If the finite groupGacts faithfully and irreducibly on the finiteGF(p)-moduleVandpdoes not divide the order ofG, is the numberk(GV) of conjugacy classes of the semidirect productGVbounded above by the order ofV? 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 thek(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 thek(G V)-problem for solvableGand large ρ.