Regular orbits of nilpotent subgroups of solvable linear groups. (Q630841)

Let \(G\) be a finite nilpotent linear group over some finite field of characteristic \(p\). Let \(V\) be the finite vector space on which \(G\) acts naturally. An element \(v\in V\) generates a \textit{regular orbit} if the only element of \(G\) which fixes \(v\) is the identity. It was proved by \textit{B. B. Hargraves}, [J. Algebra 72, 54-100 (1981; Zbl 0482.20004)], that the following is sufficient to guarantee the existence of a regular orbit in \(V\): \(G\) is completely reducible, \(G\) does not contain the wreath product \(Z_2\wr Z_2\) as a section, and, in addition, if \(p=2\) then \(G\) also does not contain the wreath product \(Z_r\wr Z_r\) as a section for \(r\) any Mersenne prime. In the present paper, this result is generalized by replacing the hypothesis that \(G\) is completely reducible by the hypothesis that there exists some finite solvable subgroup \(H\) of \(\text{GL}(V)\) such that \(O_p(H)=1\) and \(G\subseteq H\). This confirms, in particular, a conjecture of \textit{W. Carlip}, [Ill. J. Math. 38, No. 2, 198-222 (1994; Zbl 0847.20008)].