The Dade group of a metacyclic p-group
نویسندگان
چکیده
The Dade group D(P ) of a finite p-group P , formed by equivalence classes of endopermutation 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 nontrivial element of its torsion subgroup D(P ) has order 2, (or also 4, in case p = 2). The group D(P ) is closely related to the injectivity of the restriction map Res : T (P )→ ∏ E T (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.
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