Real vectors for linear groups and the \(k(GV)\)-problem
This paper may be seen as the penultimate step before reaching the solution of Brauer's \(k(B)\)-problem for a finite \(p\)-solvable group. Recalling that, in the usual notation, \(V\) is an elementary Abelian \(p\)-group on which the \(p'\)-group \(G\) acts faithfully, the authors show that the equivalent \(k(GV)\)-problem has an affirmative answer provided \(p\neq 5\). We should now note that in early 2003, the authors of the present paper and, independently, D. Gluck and K. Maagard, obtained a complete solution of the \(k(GV)\)-problem, thus settling Brauer's \(k(B)\)-problem for \(p\)-solvable groups. The main working method has been to show that \(V\) contains real vectors for \(G\), except in certain special circumstances. The analysis of minimal counterexamples \((G,V)\) that contain no real vectors has led the various investigators of the \(k(GV)\)-problem to identify six types of so-called nonreal reduced pairs \((G,V)\), which are understood in great detail. The main additional idea in the present paper is to apply results of \textit{L. G. Kovács} and \textit{G. R. Robinson} [J. Algebra 160, No. 2, 441-460 (1993; Zbl 0830.20048)] and \textit{M. W. Liebeck} and \textit{L. Pyber} [J. Algebra 198, No. 2, 538-562 (1997; Zbl 0892.20017)] concerning the number of conjugacy classes in certain permutation groups and thereby obtain the required conjugacy class estimates in \(GV\) in the absence of real vectors. The authors show great ingenuity in applying a wide variety of counting techniques, at the same time making use of information about specific configurations that arise in the course of their analysis.
scientific article; zbMATH DE number 1985881
scientific article