On generalized Schur groups

TL;DR

This paper introduces generalized Schur groups, extending the concept of Schur groups by focusing on central S-rings, and explores their properties with examples of nonabelian cases.

Contribution

It defines generalized Schur groups, generalizes the classical notion, and provides foundational properties and examples, especially for nonabelian groups.

Findings

Generalized Schur groups include all Schur groups for abelian cases.

Basic properties of generalized Schur groups are established.

Infinite families of nonabelian generalized Schur groups are constructed.

Abstract

An $S$-ring (Schur ring) is called central if it is contained in the center of the group ring. We introduce the notion of a generalized Schur group, i.e. such finite group that all central $S$-rings over this group are schurian. It generalizes in a natural way the notion of a Schur group and they are equivalent for abelian groups. We establish basic properties and provide infinite families of nonabelian generalized Schur groups