Induced characters with equal degree constituents (Q2260301)

The authors consider the following hypotheses. Let \(G\) be a finite group, and let \(H\) be a proper subgroup of \(G\) such that every nonprincipal irreducible character of \(H\) induces to a sum of irreducible characters of \(G\) all of the same degree. Using elementary arguments, they show that either \(H \subseteq G'\) or \(G' \subseteq H\), where \(G'\) is the commutator subgroup of \(G\). Using the classification of the finite simple groups, they prove that the normal closure \(H^G\) of \(H\) in \(G\) is always a proper subgroup of \(G\), and that if \(H^G\) is a proper subgroup of \(G'\), then \(H^G\) is solvable.