Some applications of a fundamental theorem by Gluck and Wolf in the character theory of finite groups (Q1084495)

First the Gluck-Wolf theorem is generalized slightly as follows: ``Let \(\pi\) be a set of primes, \(G\) a finite group, \(Z\triangleleft G\), \(G/Z\) \(\pi\)-separable and \(\lambda\in\text{Irr}(Z)\). If \(\chi(1)/\lambda(1)\) is a \(\pi'\)-number for all \(\chi\in\text{Irr}(G|\lambda)\), then \(G/Z\) has Abelian \(\pi\)-Hall subgroups.'' Then the authors give 3 applications of this result. Two of them concern the influence of the prime factorization of character degrees in a group on the group structure and the third is a proof of a ``height 0-conjecture'' for \(\pi\)-blocks in \(\pi\)-separable groups.