J. Aust. Math. Soc.  75 (2003), 193-219
A new approach to the $k(GV)$-problem

Thomas Michael Keller
  Department of Mathematics
  Texas State University
  601 University Drive
  San Marcos, TX 78666

This paper is concerned with the well-known and long-standing  k(GV)-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  p = 5  for which it is still open. In this paper we suggest a new approach to the  k(GV)-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(GV)-problem for solvable  G and large  p.
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