On nilpotent Schur groups

TL;DR

This paper classifies nonabelian nilpotent Schur groups, showing they belong to specific known families, thereby advancing understanding of their structure and properties.

Contribution

It provides a complete classification of nonabelian nilpotent Schur groups, identifying their explicit families and expanding the theory of Schur groups.

Findings

Nonabelian nilpotent Schur groups are classified into specific families.

All such groups belong to explicitly described categories.

The classification enhances understanding of Schur group structures.

Abstract

A finite group $G$ is called a Schur group if every $S$-ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of $\sym(G)$ that contains all right translations. We prove that every nonabelian nilpotent Schur group belongs to one of the explicitly given families of groups.