On isomorphisms between centers of integral group rings of finite groups

Abstract:

For finite nilpotent groups $G$ and $G’$, 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 $SG’$ is monomial, i.e., maps class sums in $SG$ to class sums in $SG’$ up to multiplication with roots of unity. As a consequence, $G$ and $G’$ have identical character tables if and only if the centers of their integral group rings $\mathbb {Z} G$ and $\mathbb {Z} G’$ are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.