A variation on theorems of Jordan and Gluck

Gluck proved that any finite group G has an abelian subgroup A such that |G:A| is bounded by a polynomial function of the largest degree of the complex irreducible characters of G. This improved on a previous bound of Isaacs and Passman. In this paper, we present a variation of this result that looks at the number of prime factors. All these results, in turn, may be seen as variations on the classical theorem of Jordan on linear groups.