On schurity of dihedral groups
Grigory Ryabov
TL;DR
This work advances the Schur ring program by analyzing the schurity of dihedral groups. It proves that any generalized dihedral Schur group must be dihedral, and derives tight arithmetic restrictions on the order $2n$ of a Schur dihedral group, notably limiting the prime divisors of $n$ to at most three and constraining their form. It then establishes a concrete infinite-family result: dihedral groups of order $2p$ are Schur when $p$ is a Fermat prime or of the form $p=4q+1$ with $q$ prime, using a combination of nonexistence results for nontrivial difference sets in $C_p$ and careful classification of $S$-rings over $D_{2p}$. The paper also provides explicit nonschurian $S$-rings arising from relative difference sets, and discusses the Schurity of related Frobenius groups and the broader landscape of nonabelian Schur groups, offering both new examples and several open questions. These results contribute to the broader goal of classifying Schur groups and illuminate deep links between $S$-rings, difference sets, and combinatorial properties of permutation 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.