On Schur p-groups of odd order

TL;DR

This paper characterizes certain odd-order p-groups called Schur groups, proving specific structures are Schur and classifying all noncyclic Schur p-groups for odd primes.

Contribution

It proves that groups of the form Z_3 × Z_{3^n} are Schur and classifies all noncyclic Schur p-groups for odd primes.

Findings

Groups Z_3 × Z_{3^n} are Schur.

All noncyclic Schur p-groups for odd p are either Z_3 × Z_3 × Z_3 or Z_3 × Z_{3^n}.

Provides a classification of certain Schur p-groups.

Abstract

A finite group $G$ is called a Schur group if any $S$-ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. We prove that the groups $\mathbb{Z}_3\times \mathbb{Z}_{3^n}$, where $n\geq 1$, are Schur. Modulo previously obtained results, it follows that every noncyclic Schur $p$-group, where $p$ is an odd prime, is isomorphic to $\mathbb{Z}_3\times \mathbb{Z}_3 \times \mathbb{Z}_3$ or $\mathbb{Z}_3\times \mathbb{Z}_{3^n}$, $n\geq 1$ .