Classification of abelian Schur groups II

TL;DR

This paper completes the classification of abelian Schur groups by proving that several nonpowerful order groups are Schur groups, expanding the known list of such groups.

Contribution

It proves that certain nonpowerful order abelian groups are Schur groups, finalizing the classification of all abelian Schur groups.

Findings

Several nonpowerful order groups are Schur groups.

The classification of abelian Schur groups is completed.

The paper extends previous classifications by confirming new 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 prove that several groups of nonpowerful order from the list are Schur groups. By that, we obtain a classification of abelian Schur groups.