So last time we proved that the dimensions of an irreducible representation divide the index of the center. Now to generalize this to an arbitrary abelian normal subgroup.

There are first a few basic background results that I need to talk about. 


Given a group {G} and a subgroup {H} (in fact, this can be generalized to a non-monomorphic map {H \rightarrow G}), a representation of {G} yields by restriction a representation of {H}. One obtains a functor {\mathrm{Res}^G_H: Rep(G) \rightarrow Rep(H)}. This functor has an adjoint, denoted by {\mathrm{Ind}_H^G: Rep(H) \rightarrow Rep(G)}. (more…)