Endo-trivial modules for finite groups with dihedral Sylow 2-subgroup

We provide a description of the torsion subgroup TT(G) of the finitely generated abelian group T(G) of endo-trivial kG-modules in the case that G has a dihedral Sylow 2-subgroup of order at least 8 and k is an algebraically closed field of characteristic p = 2. Specifically, we prove that TT(G) congruent to X(G), the group of one-dimensional kG-modules, except possibly when G/O2'(G) congruent to u(6), the alternating group of degree 6; in which case G may have nine-dimensional simple torsion endo-trivial modules. Our results complement the tame-representation type investigation of endo-trivial modules started by Carlson-Mazza-Thevenaz in the cases of semi-dihedral and generalized quaternion Sylow 2-subgroups. Furthermore, we provide a general reduction result, valid at any prime p, to recover the structure of TT(G) from the structure of TT(G/H), where H is a normal p'-subgroup of G.