Classification of abelian Schur groups I

TL;DR

This paper advances the classification of abelian Schur groups by analyzing their schurity, proving some are Schur groups and others are not, based on their structural properties.

Contribution

It completes the classification of abelian Schur groups and identifies specific groups that are or are not Schur groups.

Findings

Direct product of elementary abelian group of order 4 and certain cyclic groups is Schur.

Some groups from the list are proven to be nonschurian.

The paper refines understanding of which abelian groups are Schur groups.

Abstract

A finite group $G$ is called a Schur group if every Schur ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of the symmetric group $Sym(G)$ that contains all right translations of $G$. The list of all possible abelian Schur groups was obtained by Evdokimov, Kov\'acs, and Ponomarenko in 2016. In two papers, we complete a classification of abelian Schur groups. In the present paper, we study schurity of several groups from the list. First, we prove that a direct product of the elementary abelian group of order 4 and a cyclic group, whose order is an odd prime power or a product of two distinct odd primes, is a Schur group. Second, we establish nonschurity of some other groups from the list.