Coprime actions, fixed-point subgroups and irreducible induced characters (Q1814976)

The authors prove the following theorem: Let a finite group \(A\) act via automorphisms on a finite group \(G\), where \((|G|, |A|)=1\), and assume that some character of a subgroup \(H\subset G\) induces an irreducible character of \(G\). If \(A\) fixes every element of \(H\), then the action of \(A\) on \(G\) is trivial. Furthermore they study relations between the fixed-point indices \([H:C_H(A)]\) and \([G:C_G(A)]\) in the case where \(G\) is solvable or supersolvable.