Characterization of Frobenius Groups of Special Type

We define a Con-Cos group G to be one having a proper normal subgroup N whose cosets other than N itself are conjugacy classes. It follows easily that N = G’, the derived group of G. Most of the paper is devoted to trying to classify finite Con-Cos groups satisfying the additional requirement that N has just two conjugacy classes. We show that for such groups the center Z(G) has order at most 2, and if Z(G) = {1}, then G is a Frobenius group of a rather special type.