Signed permutation modules, Singer cycles and class numbers

Abstract

Let p be a rational prime. The k(GV) theorem states that, given a finite p′-group G acting faithfully on a finite elementary abelian p-group V, the number of conjugacy classes of the semidirect product GV is bounded above by the order of V (k(GV) ⩽ |V|). In the present paper we examine when the upper bound k(GV) = |V| is attained. It is shown that for p > 5 this happens if and only if G/CG(U) ≌ CG(V/U) is cyclic of order |U| – 1 for each nontrivial irreducible submodule U of V (Singer cycle). It remains open whether this is also true when p = 5. For p = 2, 3 there exist examples where equality holds but G is not abelian.