The Dade group of a metacyclic p-group

Abstract

The Dade group D( P) of a finite p-group P, formed by equivalence classes of endo-permutation modules, is a finitely generated Abelian group. Its torsion-free rank equals the number of conjugacy classes of non-cyclic subgroups of P and it is conjectured that every non-trivial element of its torsion subgroup D t ( P) has order 2, (or also 4, in case p=2). The group D t ( P) is closely related to the injectivity of the restriction map Res :T(P)→∏ ET(E) , where E runs over elementary Abelian subgroups of P and T( P) denotes the group of equivalence classes of endo-trivial modules, which is still unknown for (almost) extra-special groups ( p odd). As metacyclic p-groups have no (almost) extra-special section, we can verify the above conjecture in this case. Finally, we compute the whole Dade group of a metacyclic p-group.