A variation on a theorem of Gluck

Abstract

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. Moreto (J. Algebra 301:274–279, 2006) presented a variation of this result that looks at the number of prime factors and obtained an almost quadratic bound. In this note, we improve the result of Moreto to almost linear.