Projective Modules and Involutions

Let G be a finite group, and let Ω := {t ∈ G | t = 1}. Then Ω 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 kΩ 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 kΩ, establishes a bijection between the projective components of kΩ and the real 2-blocks of G of defect zero. Let G be a finite group, with identity element e, and let Ω := {t ∈ G | t = e}. Then Ω is a G-set under conjugation. In this note we describe the projective components of the permutation module kΩ, where k is an algebraically closed field of characteristic 2. By a projective component we mean an indecomposable direct summand of kΩ that is also a direct summand of a free kG-module. We show that all such components are irreducible, self-dual and occur with multiplicity 1. This gives an alternative proof of Remark (2) on p. 254 of [5], and strengthens Corollaries 3 through 7 of that paper. In addition, we can give the following quick proof of Proposition 8 in [5]: Corollary 1. Suppose that H is a strongly embedded subgroup of G. Then kH↑ G ∼= kG⊕[⊕ s i=1Pi] where s ≥ 0 and the Pi are pairwise nonisomorphic self-dual projective irreducible kG-modules. Proof. That H is strongly embedded means that |H | is even and |H ∩H| is odd, for each g ∈ G\H . Let t ∈ H be an involution. Then clearly CG(t) ≤ H . So kH↑ G is isomorphic to a submodule of (kCG(t))↑ . Mackey’s theorem implies that every component of kH↑ , other than kG, is a projective kG-module. Being projective, these modules must be components of (kCG(t))↑ . The result now follows from Theorem 8. Consider the wreath product G ≀ Σ of G with a cyclic group Σ of order 2. Here Σ is generated by an involution σ and G ≀Σ is isomorphic to the semidirect product of the base group G×G by Σ. The conjugation action of σ on G ×G is given by (g1, g2) σ = (g2, g1), for all g1, g2 ∈ G. The elements of G ≀Σ will be written (g1, g2), (g1, g2)σ or σ. We shall exploit the fact that kG is a kG ≀Σ-module. For, as is well-known, kG is an k(G×G)-module via: x · (g1, g2) := g −1 1 xg2, for each x ∈ kG, and g1, g2 ∈ G. The action of Σ on kG is induced by the permutation action of σ on the Date: March 11, 2004. 1991 Mathematics Subject Classification. 20C20.