The Dade group of a metacyclic \(p\)-group
Let \(P\) be a finite \(p\)-group and \(k\) an algebraically closed field of characteristic \(p\). A \(kP\)-module \(M\) is endo-permutation, if \(\text{End}_k(M)\) is a permutation \(kP\)-module. In this case \(M\) has a unique (up to isomorphism) indecomposable summand \(M_0\) with vertex \(P\), called the cap of \(M\). Two endo-permutation modules are called equivalent if they have isomorphic caps. The equivalence classes of endo-permutation \(kP\)-modules form an Abelian group \(D(P)\) with respect to the tensor product over \(k\) and taking the \(k\)-dual. The main result of this paper is the computation of \(D(P)\) when \(P\) is metacyclic. It is also shown that in this case, the torsion subgroup of \(D(P)\) has order \(2\) (or also \(4\) in the case \(p=2\)).
scientific article; zbMATH DE number 1967901