Relatively closed subgroups of permutation groups with a cyclic regular normal subgroup
Alexander Buturlakin, Andrey V. Vasil'ev
TL;DR
The paper develops a comprehensive framework for relatively closed subgroups of permutation groups with a cyclic regular normal subgroup $W$ and a cyclic point stabilizer $A$, focusing on one-dimensional affine groups. It introduces a normal form for subgroups, analyzes orbit structures of cyclic subgroups, and defines the radical $ ext{rad}(H)$ to obtain a precise arithmetic criterion: $H$ is relatively closed in $G$ iff $ ext{rad}(H)=Higcap W$ and $C_A(W/(Higcap W)) leq H$, with the relative closure given by $H^{ ext{rel}}=H ext{rad}(H) C_A(W/ ext{rad}(H))$. The authors classify maximal and second-maximal relatively closed subgroups, building a lattice that enables recursive description of all relatively closed subgroups, and apply these results to one-dimensional affine groups and their schemes, yielding a generalization of Muzychuk’s rank-3 results and enabling a systematic study of rank-4 affine schemes. The work connects group-closure phenomena to association schemes and to concrete graphs (e.g., Paley and Peisert), providing structural tools with implications for the classification of one-dimensional affine schemes and affine-rank groups. Overall, the paper unifies algebraic and combinatorial perspectives to advance the understanding of relational structures arising from affine permutation groups.
Abstract
Motivated by some known problems concerning combinatorial structures associated with finite one-dimensional affine permutation groups, we study subgroups which are closed in $\operatorname{Γ{L}}_1(q)$. 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.