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

Abstract

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 ) ⊗ Z D ( 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 ⊗ Z D ( 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 ) .