Finite group p-modular representation theory of a finite group with a non-trivial central p'-subgroup

Let p be a prime integer, let G be a finite group with a non-trivial p'-subgroup Z of Z(G). Let k be a field of prime characteristic p that is a splitting field for all subgroups of G. Here, by multiplication, Z permutes the p'-conjugacy classes of G. We show how these orbits of Z yield a classification of the isomorphism types of simple kG-module. This analysis utilizes and extends a celebrated theorem of R. Brauer when Z = 1 . Then we apply this analysis to the groups G = SL(2, q) where q is a power of p, p ? 2 and Z = Z(G). We prove that kG has q-1/2 faithful and 1+ q-1/2 non faithful isomorphism types of simple modules.