Projective modules and involutions

Let G be a finite group, and let Omega := {t is an element of G vertical bar t(2) = 1}. Then Q is a G-set under conjugation. Let k be an algebraically closed field of characteristic 2. It is shown that each projective indecomposable summand of the G-permutation module k Omega is irreducible and self-dual, whence it belongs to a real 2-block of defect zero. This, together with the fact that each irreducible kG-module that belongs to a real 2-block of defect zero occurs with multiplicity I as a direct summand of k Omega, establishes a bijection between the projective components of k Omega and the real 2-blocks of G of defect zero. (c) 2005 Elsevier Inc. All rights reserved.