Centralizers and character degrees

Abstract

Let the finite solvable group A be acting on a finite group G and suppose that (|A|,|G|) = 1. The Glauberman correspondence provides a natural one-to-one correspondence between the irreducible characters of G invariant under A and all the irreducible characters of C G (A). Let cd(C G (A)) denote the set of degrees of the irreducible characters of C G (A) and let cd A (G) denote the set of degrees of the A-invariant irreducible characters of G. We prove that for every pair of positive integers (n, m), if m > 1, then there exists A and G as above such that \(|{\rm cd}({\rm C}_G(A))| = n\) and \( |{\rm cd}_A(G)| = m\).