Abelian Schur groups of odd order

TL;DR

This paper classifies abelian Schur groups of odd order, proving that groups of the form C_3×C_3×C_p are Schur for any prime p, completing the classification.

Contribution

It provides a complete classification of abelian Schur groups of odd order, identifying a new family of such groups involving C_3×C_3×C_p.

Findings

C_3×C_3×C_p is Schur for any prime p

Completes the classification of abelian Schur groups of odd order

Extends previous results on Schur groups

Abstract

A finite group $G$ is called a Schur group if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. It is proved that the group $C_3\times C_3\times C_p$ is Schur for any prime $p$. Together with earlier results, this completes a classification of the abelian Schur groups of odd order.