Finite groups with almost distinct character degrees

Finite groups with the nonlinear irreducible characters of distinct degrees, were classified by the authors and Berkovich. These groups are clearly of even order. In groups of odd order, every irreducible character degree occurs at least twice. In this article we classify finite nonperfect groups G, such that chi (1) = theta (1) if and only if theta = 7 for any nonlinear chi not equal theta is an element of Irr(G). We also present a description of finite groups in which xG' subset of class(x) boolean OR class(x(-1)) for every x is an element of G - G'. These groups generalize the Frobenius groups with an abelian complement, and their description is needed for the proof of the above mentioned result on characters. (C) 2006 Elsevier Inc. All rights reserved.