Groups with a Fixed Character Degree: The General Case

We obtain arithmetic conditions which are satisfied by solvable groups admitting an abelian $π$-subgroup. First, we extend a previous characterization by removing the square-free hypothesis on some fixed irreducible character degree. We then show the same arithmetic conditions follow from a faithful action of an abelian $π$-subgroup on the $π'$-part of the Fitting subgroup, where $π$ is the set of prime divisors of the aforementioned character degree.