On the isomorphism problem for the ring of monomial representations of a finite group

In this paper we are concerned with the problem of finding properties of a finite group G in the ring D(G) of monomial representations of G. We determine the conductors of the primitive idempotents of Q(ζ)⊗ZD(G), where ζ∈C is a primitive |G|-th root of unity, and prove a structure theorem for the torsion units of D(G). Using these results we show that an abelian group G is uniquely determined by the ring D(G). We state an explicit formula for the primitive idempotents of Z[ζ]p⊗ZD(G), where Z[ζ]p is a localization of Z[ζ]. We get further results for nilpotent and p-nilpotent groups and we obtain properties of Sylow subgroups of G from D(G).