Large orbits of subgroups of solvable linear groups.
Let \(G\) be a finite group and \(V\) a finite, faithful and completely reducible \(G\)-module. It is a classical theme to study orbit structures of \(G \) acting on \(V\). One of the most important and natural questions about orbit structure is to establish the existence of an orbit of a certain size. In this paper, the author proves the following results. Suppose that \(G\) is a finite solvable group and \(V\) is a finite, faithful and completely reducible \(G \)-module. Let \(H\) be a nilpotent subgroup of \(G\); then \(H\) has at least three regular orbits on \(V\oplus V\). Let \(H\) be a subgroup of \(G\) and \(3\nmid |H|\); then \(H\) has at least three regular orbits on \(V\oplus V\). Let \( H\) be a subgroup of \(G\) and assume the Sylow \(2\)-subgroups of the semidirect product \(HV\) are Abelian; then \(H\) has at least two regular orbits on \(V\oplus V\).
scientific article; zbMATH DE number 6308384