Existence of non-perfect-code maximal subgroups of symmetric groups

Determine whether there exists a maximal subgroup A of the symmetric group S_n that is not a perfect code of S_n.

Background

The paper studies subgroup perfect codes in finite groups and, more generally, perfect codes of pairs (G,H) associated with vertex-transitive coset graphs. After proving that every subgroup S_m is a perfect code of S_n for the natural embedding S_m<S_n, the authors restrict attention to maximal subgroups of S_n, since these form a broad and well-studied class and larger subgroup perfect codes correspond to sparser underlying Cayley graphs.

Computational experiments using GAP show that every maximal subgroup of S_n is a perfect code of S_n for n≤9. The paper proves the result for all intransitive maximal subgroups and establishes a partial result for affine maximal subgroups, but it does not determine whether exceptional maximal subgroups that fail to be perfect codes exist in general. Thus the unresolved problem is to decide the existence of any maximal subgroup A of S_n that is not a perfect code of S_n.