Structure of normal subgroups with three G-class sizes

Let G be a finite group and N be a normal subgroup of G. Suppose that the set of G-conjugacy class sizes of N is {1, m, n}, with m < n and m does not divide n. In this paper, we show that N is solvable, and we determine the structure of these subgroups.
A. Beltran and M. J. Felipe are supported by Proyecto MTM2007-68010-C03-03, by Proyecto MTM2010-19938-C03-02 and by Proyecto GV-2009-021. A. Beltran is also supported by grant Fundacio Caixa-Castello P11B2008-09.