Remarks on the existence of regular orbits of finite operator groups (Q1307047)

As to so-called Fitting length problems, if one knows that a certain operator group acts on a vector space admitting regular orbits. In recent years a lot of research has been done around this theme. The author of this paper pursues on papers of \textit{P. Fleischmann} [J. Algebra 103, 211-215 (1986; Zbl 0604.20014)], \textit{D. Gluck} [Can. J. Math. 35, 59-67 (1983; Zbl 0509.20002)], and others, especially \textit{A. Turull} [J. Reine Angew. Math. 371, 67-91 (1986; Zbl 0587.20017); Math. Proc. Camb. Philos. Soc. 107, No. 2, 227-238 (1990; Zbl 0704.20016)]. Let us mention one result obtained here. Suppose \(A\) is a finite nilpotent group with \(B\in\text{Syl}_2(A)\) with no section isomorphic to a dihedral group of order 8; let \(K\) be a field with \(\text{Char}(K)=p\geq 3\). If \((p,|A|)=1\), then there is a regular \(A\)-orbit on \(V\) for every \(KA\)-module \(V\). If \(V\neq\{0\}\), then there are at least two regular \(A\)-orbits on \(V\). Reviewer's remark: It seems interesting to compare the results as obtained to those many of the authors Turull (again), Espuelas, Berger, Hargraves, to mention but a few.