The character degree ratio and large abelian subgroups

Abstract

Let b(G) be the largest degree of the irreducible characters of a nonabelian finite group G and let m(G) be the smallest degree of the nonlinear irreducible characters of G. M. Isaacs defined the character degree ratio of G to be rat(G)=b(G)/m(G). He proved that for nonabelian solvable groups the derived length is bounded by a logarithmic function of the character degree ratio. In a similar spirit, results that restrict the structure of nonsolvable groups in terms of the character degree ratio were later proved by J. P. Cossey, M. Lewis and H. N. Nguyen. In this note we show that, with some rather precisely determined exceptions, G has an abelian subgroup of index bounded by a polynomial function in rat(G). As a consequence we get stronger (in most situations) versions of most of the previously known results on the character degree ratio.