Isomorphic rings of monomial representations

Let D(G) and be the rings of monomial representations of finite groups G and of odd order. We prove that the isomorphy implies the isomorphy of the respective Burnside rings. Moreover we prove that the isomorphy with finite Z-groups G and (i.e. groups which have only cyclic Sylow subgroups) implies the isomorphy . In particular a group G of squarefree order is uniquely determined up to isomorphy by the ring D(G).