On group actions of p‐groups on p′$p^{\prime }$‐groups

Abstract

AbstractLet A be a finite group that acts coprimely on another finite group G, and in this setting we study the orbit structure of the group A on the irreducible characters or conjugacy classes of the group G. We prove that if a p‐group A acts coprimely on a finite group G, then A has a “large” orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This answers a conjecture of Moretó affirmatively.