On schurity of dihedral groups | Sciact

On schurity of dihedral groups Full article

Journal

Journal of Algebra
ISSN: 0021-8693 , E-ISSN: 1090-266X

Output data

Year: 2025, Volume: 682, Pages: 247-277 Pages count : 31 DOI: 10.1016/j.jalgebra.2025.05.035

Tags

Schur rings, Schur groups, Difference sets, Dihedral groups

Authors

Ryabov Grigory ^1,2

Affiliations

1	School of Mathematical Sciences, Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University
2	Novosibirsk State Technical University,

A finite group G is called a Schur group if every S-ring over Gis 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 qis also prime, is Schur. Towards this result, we prove nonexistence of a difference set in a cyclic group of order p ̸=13and classify all S-rings over some dihedral groups.

Ryabov G.
On schurity of dihedral groups
Journal of Algebra. 2025. V.682. P.247-277. DOI: 10.1016/j.jalgebra.2025.05.035 WOS Scopus OpenAlex

Submitted:	Feb 19, 2025
Published online:	Jun 17, 2025

Web of science:	WOS:001518652600006
Scopus:	2-s2.0-105008540757
OpenAlex:	W4411371978

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