On schurity of 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.