Separability of Schur rings over abelian groups of odd order

TL;DR

This paper proves that all abelian groups of order 9p are separable with respect to all finite abelian groups, completing a classification and linking to graph isomorphism problems.

Contribution

It establishes the separability of abelian groups of order 9p, extending previous results and completing the classification for odd order abelian groups.

Findings

All abelian groups of order 9p are separable with respect to all finite abelian groups.

This classification completes the understanding of separability for noncyclic abelian groups of odd order.

The Weisfeiler-Leman dimension for Cayley graphs over these groups is at most 2.

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.