Quasiprimitive groups with a biregular dihedral subgroup,and arc-transitive bidihedrants

TL;DR

This paper classifies quasiprimitive permutation groups with a biregular dihedral subgroup and characterizes arc-transitive graphs with automorphism groups containing such subgroups, revealing their structure as normal covers of simpler arc-transitive graphs.

Contribution

It provides a classification of quasiprimitive groups with biregular dihedral subgroups and characterizes related arc-transitive graphs, advancing understanding of their symmetry properties.

Findings

Every such graph is a normal cover of an arc-transitive graph with specific automorphism group properties.

Complete classification of quasiprimitive or bi-quasiprimitive arc-transitive graphs with these subgroups.

Identification of the structure of automorphism groups in these graphs.

Abstract

A semiregular permutation group on a set $\Ome$ is called {\em bi-regular} if it has two orbits. A classification is given of quasiprimitive permutation groups with a biregular dihedral subgroup. This is then used to characterize the family of arc-transitive graphs whose automorphism groups containing a bi-regular dihedral subgroup. We first show that every such graph is a normal $r$-cover of an arc-transitive graph whose automorphism group is either quasiprimitive or bi-quasiprimitive on its vertices, and then classify all such quasiprimitive or bi-quasiprimitive arc-transitive graphs.