Centralizers and character degrees (Q1581194)

Let a finite \(\pi'\)-group \(A\) act on finite \(\pi\)-group \(G\). We denote by \(\text{cd}_A(G)\) the set of degrees of the \(A\)-invariant irreducible characters of \(G\). Set \(C=C_G(A)\). A natural question is whether one could use the Glauberman-Isaacs correspondence to obtain information on \(|\text{cd}(C)|\) from \(|\text{cd}_A(G)|\), where \(\text{cd}(C)\) is the set of degrees of irreducible characters of \(C\). The following result is proved: Theorem. Let \(\{n,m\}\) be any pair of positive integers, subject only to the condition that \(m>1\). Then there exist finite groups \(A\) and \(G\), with \(A\) acting on \(G\) by automorphisms and \((|A|,|G|)=1\) such that \(|\text{cd}_A(G)|=n\) and \(|\text{cd}(G)|=m\).