Permutation groups, pattern involvement, and Galois connections

There is a connection between permutation groups and permutation patterns: for any subgroup $G$ of the symmetric group $S_{\mathrm{\ell }}$ and for any $n\ge \mathrm{\ell }$, the set of $n$-permutations involving only members of $G$ as $\mathrm{\ell }$-patterns is a subgroup of $S_n$. Making use of the monotone Galois connection induced by the pattern avoidance relation, we characterize the permutation groups that arise via pattern avoidance as automorphism groups of relations of a certain special form. We also investigate a related monotone Galois connection for permutation groups and describe its closed sets and kernels as automorphism groups of relations.