Abstract

Let G be a nite group. In this paper we investigate the permutation module of G acting by conjugation on its involutions, over a eld of characteristic 2. This develops the main theme of [10] and [8]. In the former paper G. R. Robinson considered the projective components of this module. In the latter paper the author showed that each such component is irreducible and self-dual and belongs to a 2-blocks of defect zero. Here we investigate which 2-blocks have a composition factor in the involution module. There are two apparently dierent ways of characterising such blocks. One method is local and uses the defect classes of the block. This gives rise to the denition of a strongly real 2-block. The other method is global and uses the Frobenius-Schur indicators of the irreducible characters in the block. Our main result is Theorem 2. The proof of this theorem requires Corollaries 4, 15, 18 and 20.