Regular orbits of self-dual modules (Q2031458)

The authors prove that if $V$ is a faithful completely reducible self-dual $FG$-module for a finite solvable group $G$, where $F$ is a finite field of characteristic $p$, then if either $p$ is odd, or $p = 2$ and $|H|$ is not divisible by 3, then the subgroup $H$ of $G$ has at least two regular orbits on $V$. This improves earlier results of a similar nature by \textit{A. Espuelas} [J. Algebra 138, No. 1, 1--12 (1991; Zbl 0654.20004)] and by the second author [J. Algebra 351, No. 1, 220--234 (2012; Zbl 1252.20008)].