Isomorphisms between centers of integral group rings

Abstract

For finite nilpotent groups $G$ and $G^{\prime }$, and a $G$-adapted ring $S$ (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings $SG$ and $S{G}^{\prime }$ is monomial, i.e., maps class sums in $SG$ to class sums in $S{G}^{\prime }$ up to multiplication with roots of unity. As a consequence, $G$ and $G^{\prime }$ have identical character tables if and only if the centers of their integral group rings $\mathbb{Z}G$ and $\mathbb{Z}{G}^{\prime }$ are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

11 pages