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_ℓ and for any n ≥ ℓ, the set of n-permutations involving only members of G as ℓ-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.