Fixed Points and Characters in Groups with Non-Coprime Operator Groups

Abstract

Let A and H be finite groups with A acting on H, i.e., there is a given, fixed homomorphism A → Aut(H). In this situation, A acts on the set of conjugacy classes of H and also on the set of irreducible characters of H. If A is cyclic, it follows from a lemma of Brauer (see, for instance, 1, 12.1) that the number of fixed points in these two actions are equal and therefore one can conclude that for any A, the number of orbits in the two actions are equal.