The Dade group of a metacyclic p-group

✍ Scribed by Nadia Mazza

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 115 KB
• Volume: 266
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00328-4

✦ Synopsis

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 ) → 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.