On Schur Rings Over a Semi Dihedral Group

TL;DR

This paper proves that the only primitive Schur ring over a semi dihedral group is trivial and establishes that semi dihedral subgroups are Burnside groups, implying certain transitivity properties.

Contribution

It demonstrates the uniqueness of the trivial Schur ring over semi dihedral groups and links semi dihedral subgroups to Burnside groups, revealing new structural insights.

Findings

Only trivial primitive Schur ring exists over semi dihedral groups

Semi dihedral subgroups are Burnside groups

Primitive groups containing semi dihedral subgroups are 2-transitive

Abstract

In this paper, we shall show that the only primitive Schur ring over a semi dihedral group is the trivial one and every semi dihedral subgroup is Burnside group, that is a primitive group containing a regular subgroup isomorphic to the semi dihedral subgroup is necessarily 2-transitive.