Modular representation theory of finite groups with T.I. Sylow 𝑝-subgroups

Abstract

Let p p be a fixed prime, and let G G be a finite group with a T.I. Sylow p p -subgroup P P . Let N = N G ( P ) N = {N_G}(P) and let k ( G ) k(G) be the number of conjugacy classes of G G . If z ( G ) z(G) denotes the number of p p -blocks of defect zero, then we show in this article that z ( G ) = k ( G ) − k ( N ) z(G) = k(G) - k(N) . This result confirms a conjecture of J. L. Alperin. Its proof depends on the classification of the finite simple groups. Brauer’s height zero conjecture and the Alperin-McKay conjecture are also verified for finite groups with a T.I. Sylow p p -subgroup.