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.