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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.

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