Direct products of groups and regular orbits (Q1680279)

Let \(G\) be a finite group acting faithfully on a module \(V\). A regular orbit of \(G\) on \(V\) is any \(G\)-orbit in \(V\) of size \(| G |\), hence \(v \in V\) is in a regular orbit if and only if \(C_{G}(v)=1\). If \(G\) is abelian, then \(V\) always has regular orbits. When \(G\) is nilpotent, this is no longer the case and the issue of determining a condition that will guarantee the existence of regular orbits was studied by \textit{B. Hargraves} [J. Algebra 72, 54--100 (1981; Zbl 0482.20004)]. It is a consequence of Hargraves' results that if \(G\) is nilpotent and two or more Sylow subgroups of \(G\) are not cyclic, then \(G\) has a regular orbit on \(V\). In this paper, the authors generalize this result to arbitrary finite \(p\)-solvable groups. The main result is the following Theorem: Suppose that \(V\) is a faithful irreducible \(KG\)-module where \(K\) is a field of characteristic \(p \geq 0\), and \(G\) is a finite group that is the direct product \(G = G_{1} \times G_{2}\) of two non-cyclic subgroups such that \((| G_{1} | , | G_{2}|)=1\). If \(p > 0\), assume that \(G\) is \(p\)-solvable. Then, \(G\) has a regular orbit on \(V\). Examples prove that this result is the best possible in the following sense. It is not possible to replace the condition that \(G_{1}\) and \(G_{2}\) are non-cyclic by the condition that \(G_{1}\) and \(G_{2}\) are non-trivial and it is not even possible remove the condition that \(G_{1}\) and \(G_{2}\) are of coprime order.