On the isomorphism problem for the ring of monomial representations of a finite group
For a finite group \(G\), the monomial representation ring \(D(G)\) is defined as the Grothendieck ring of all monomial representations of \(G\); it is an analog of the Burnside ring \(B(G)\) which is the Grothendieck ring of all permutation representations of \(G\). Moreover, \(B(G)\) can be viewed as a subring of \(D(G)\). The ring \(D(G)\) was investigated by Dress, Boltje and others. The paper under review continues these investigations. Its main results are concerned with torsion units in \(D(G)\), and with conductors of idempotents in \(\mathbb{Q}(\zeta)\otimes_\mathbb{Z} D(G)\) and \(\mathbb{Z}[\zeta]_{\mathfrak{p}} \otimes_{\mathbb{Z}} D(G)\); here \(\zeta\) is a primitive \(\left|G\right|\)-th root of unity in \(\mathbb{C}\), and \(\mathfrak{p}\) is a maximal ideal in \(\mathbb{Z}[\zeta]\). The author shows, as an application, that \(D(G)\) determines the order, the solvability, the nilpotency and the commutativity of \(G\). Furthermore, \(D(G)\) detects the commutativity of the Sylow \(p\)-subgroups of \(G\), for a prime number \(p\). It also determines whether \(O_p(G) = 1\). For an abelian group, \(D(G)\) even determines the isomorphism type of \(G\). These results are analogs of similar results for the Burnside ring \(B(G)\). Reviewer's remark: It seems to be difficult to construct nonisomorphic finite groups \(G, H\) with isomorphic rings \(D(G),~D(H)\).
scientific article; zbMATH DE number 5987872