Coprime actions and degrees of primitive inducers of invariant characters (Q2758194)

Let \(A\) and \(G\) be finite groups of coprime orders such that \(A\) acts on \(G\), and let \(\chi\) be an \(A\)-invariant irreducible complex character of \(G\). A result of \textit{I.~M.~Isaacs, M.~L.~Lewis} and \textit{G.~Navarro} [Arch. Math. 74, No. 6, 401-409 (2000; Zbl 0967.20004)] states that the degrees of any two \(A\)-primitive characters of \(A\)-invariant subgroups inducing \(\chi\) coincide.NEWLINENEWLINENEWLINEThe authors show that this result does not extend to supersolvable groups in general, but it extends if the degree of \(\chi\) is a prime power. They also show that the result extends to the case when \(G\) has a normal subgroup \(N\) such that \(G/N\) is nilpotent and \(N\) has Abelian Sylow subgroups.