On the existence of regular orbits (Q1209037)

Let \(G\) be a finite group and \(V\) a finite dimensional \(FG\)-module. When \(G\) has a regular orbit on \(V\) is an interesting problem in the theory of finite groups. In this note we first reveal connections between the structure of the finite group \(G\) and the field \(F\), where \(G\) has no regular orbits on \(V\). We then investigate the number problem of regular orbits of a finite \(D_ 8\)-free 2-group. We will be concerned only with finite 2-groups. Theorem 1: Suppose that \(G\) is a finite 2-group and that \(V\) is a faithful irreducible \(FG\)-module of finite dimension, where \(F = GF(q)\), the Galois field of \(q\) elements, and \(q\) is an odd prime. Then \(G\) has at least one regular orbit on \(V\) unless \(q = 2^ m + 1\) \((\geq 5)\) or \(2^ m - 1\) and \(G\) contains the dihedral group \(D_{2^{m + 1}}\) of order \(2^{m + 1}\) as a section.