Finite groups with almost distinct character degrees

AbstractFinite 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 χ(1)=θ(1) if and only if θ=χ¯ for any nonlinear χ≠θ∈Irr(G). We also present a description of finite groups in which xG′⊆class(x)∪class(x−1) for every x∈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