On schurity of dihedral groups

TL;DR

This paper investigates the schurity of dihedral groups, establishing conditions under which they are Schur and identifying specific orders where dihedral groups are Schur, contributing to the understanding of nonabelian Schur groups.

Contribution

It proves that generalized dihedral Schur groups are dihedral, provides necessary conditions for dihedral groups to be Schur, and classifies all S-rings over certain dihedral groups.

Findings

Dihedral groups of order 2p with p a Fermat prime or of the form 4q+1 are Schur.

Nonexistence of difference sets in cyclic groups of order p ≠ 13.

Classification of all S-rings over some dihedral groups.

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. One of the crucial questions in the $S$-ring theory is the question on schurity of nonabelian groups, in particular, on existence of an infinite family of nonabelian Schur groups. In this paper, we study schurity of dihedral groups. We show that any generalized dihedral Schur group is dihedral and obtain necessary conditions of schurity for dihedral groups. Further, we prove that a dihedral group of order $2p$, where $p$ is a Fermat prime or prime of the form $p=4q+1$, where $q$ is also prime, is Schur. Towards this result, we prove nonexistence of a difference set in a cyclic group of order $p\neq 13$ and classify all $S$-rings over some dihedral groups.