Characterization of cyclic Schur groups

TL;DR

This paper characterizes which cyclic groups are Schur groups, showing they belong to specific families of integers involving prime powers and products of distinct primes.

Contribution

It extends the classification of Schur groups to all cyclic groups, identifying precise conditions based on their order.

Findings

Cyclic groups of order p^k, pq^k, 2pq^k, pqr, 2pqr are Schur groups.

A cyclic p-group is Schur if and only if p ≥ 5.

The classification covers all cyclic groups based on their order's prime factorization.

Abstract

A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a permutation group on the set $G$ containing the regular subgroup of all right translations. It was proved by R. P\"oschel (1974) that given a prime $p\ge 5$ a $p$-group is Schur if and only if it is cyclic. We prove that a cyclic group of order $n$ is a Schur group if and only if $n$ belongs to one of the following five (partially overlapped) families of integers: $p^k$, $pq^k$, $2pq^k$, $pqr$, $2pqr$ where $p,q,r$ are distinct primes, and $k\ge 0$ is an integer.