The mod p K-theory of classifying spaces of finite groups (Q579670)

Let G be a finite group. Let p be a prime and P be a p-Sylow subgroup of G. The inclusion of P into G induces a restriction map of representation rings, r: R(G)\(\to R(P)\). The author proves there is a natural isomorphism of rings \(\hat R(G)\otimes {\mathbb{Z}}/p\simeq Image(r\otimes {\mathbb{Z}}/p).\) The \({\mathbb{Z}}/p\)-vector space dimension of this algebra equals the number of conjugacy classes of G represented by elements of order a power of p.