The Grothendieck ring of invertible modules over nilpotent groups
For a finite group \(G\), a permutation \(ZG\)-module \(P\) is defined as a \(ZG\)-lattice with a \(Z\)-basis permuted by \(G\), and the direct summands of such \(P\) are called invertible \(ZG\)-modules. Let \({\mathcal L}(QG)\) denote the Grothendieck ring of \(QG\)-modules \(V\) containing a full invertible \(ZG\)-lattice. The author considers \({\mathcal L}(QG)\) in the case that \(G\) is nilpotent of odd order. Then \({\mathcal L}(QG)\) is the sum of the images of \(i_ *: {\mathcal L}(QH)\to{\mathcal L}(QG)\) for all \(p\)-elementary subgroups \(i:H\hookrightarrow G\). As a main result, the author determines the \(QH\)- modules containing a full invertible \(ZH\)-lattice, which yields a particular set of generators of \({\mathcal L}(QG)\).
scientific article; zbMATH DE number 571353