Separability of Schur rings over abelian groups of odd order

Abstract

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. A finite group $G$ is said to be separable with respect to $\mathcal{K}$ if every $S$-ring over $G$ is separable with respect to $\mathcal{K}$. We prove that every abelian group $G$ of order $9p$, where $p$ is a prime, is separable with respect to the class of all finite abelian groups. Modulo previously obtained results, this completes a classification of noncyclic abelian groups of odd order that are separable with respect to the class of all finite abelian groups. Also this implies that the Weisfeiler-Leman dimension of the class of Cayley graphs over $G$ is at most 2.