On schurity of finite abelian groups

TL;DR

This paper classifies abelian Schur groups, showing they belong to specific families, with detailed descriptions for odd order groups and proving certain groups are Schur groups.

Contribution

It provides a complete classification of abelian Schur groups, extending previous results on cyclic groups and identifying new families.

Findings

Any non-cyclic abelian Schur group of odd order is isomorphic to Z_3×Z_{3^k} or Z_3×Z_3×Z_p.

Z_2×Z_2×Z_p is a Schur group for every prime p.

The paper explicitly describes all abelian 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. Recently, the authors have completely identified the cyclic Schur groups. In this paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to $Z_3\times Z_{3^k}$ or $Z_3\times Z_3\times Z_p$ where $k\ge 1$ and $p$ is a prime. In addition, we prove that $Z_2\times Z_2\times Z_p$ is a Schur group for every prime $p$.