Endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups

Abstract

We study the finitely generated abelian group T (G) of endo-trivial kG-modules where kG is the group algebra of a finite group G over a field of characteristic $${p > 0}$$ . When the representation type of the group algebra is not wild, the group structure of T (G) is known for the cases where a Sylow p-subgroup P of G is cyclic, semi-dihedral and generalized quaternion. We investigate T (G), and more accurately, its torsion subgroup TT(G) for the case where P is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involutions under which $${TT(G) = f^{-1}(X(N_{G}(P)))}$$ holds, where $${f^{-1}(X(N_{G}(P)))}$$ denotes the abelian group consisting of the kG-Green correspondents of the one-dimensional kN G (P)-modules. We show that the lift to characteristic zero of any indecomposable module in TT(G) affords an irreducible ordinary character. Furthermore, we show that the property of a module in f −1(X(N G (P))) of being endo-trivial is not intrinsic to the module itself but is decided at the level of the block to which it belongs.