Orders generated by character values

Let K: = Q(G) be the number field generated by the complex character values of a finite group G. Let Z K be the ring of integers of K. In this paper we investigate the suborder Z[G] of Z K generated by the character values of G. We prove that every prime divisor of the order of the finite abelian group Z K/ Z[G] divides |G|. Moreover, if G is nilpotent, we show that the exponent of Z K/ Z[G] is a proper divisor of |G| unless G= 1. We conjecture that this holds for arbitrary finite groups G.