Large orbits of odd-order subgroups of solvable linear groups.
Let \(G\) be a finite solvable group and \(V\) a finite, faithful, completely reducible \(G\)-module. If \(H\) is an odd-order subgroup of \(G\), then \(H\) has at least two regular orbits on \(V\oplus V\). If, in addition \(|V|\) is odd, then there exists \(v\in V\) such that \(C_H(v)\subseteq F_2(G)\) (Theorem 1.2). Assume that, under the same hypotheses, \(V\) is endowed with a nonsingular symplectic form fixed by \(G\), then \(H\) has at least two regular orbits on \(V\) (Theorem 1.3).
scientific article; zbMATH DE number 6046974
scientific article