Abstract

Let $G$ be a finite group, and let $\Omega:=\{t\in G\mid t^2=1\}$. Then $\Omega$ 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 $1$ 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.