J. Aust. Math. Soc.
75 (2003), 193-219
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A new approach to the
 -problem
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Thomas Michael Keller
Department of Mathematics
Texas State University
601 University Drive
San Marcos, TX 78666
USA
keller@swt.edu
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Abstract
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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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