Relatively closed subgroups of permutation groups with a cyclic regular normal subgroup

TL;DR

This paper classifies relatively closed subgroups of permutation groups with a cyclic regular normal subgroup, extending the understanding of affine permutation groups and their associated combinatorial structures.

Contribution

It provides a comprehensive classification of these subgroups, generalizing recent results on affine association schemes and rank 3 graphs.

Findings

Classification of relatively closed subgroups with cyclic regular normal subgroups

Extension of Muzychuk's classification to affine association schemes

Description of minimal nontrivial one-dimensional affine schemes

Abstract

Motivated by some known problems concerning combinatorial structures associated with finite one-dimensional affine permutation groups, we study subgroups which are closed in $\operatorname{\Gamma{L}}_1(q)$. This brings us to a description of the relatively closed subgroups of permutation groups with a cyclic regular normal subgroup. Our results, in particular, provide a classification of the minimal nontrivial one-dimensional affine association schemes which generalizes the recent Muzychuk classification of the one-dimensional affine rank 3 graphs.