For an arbitrary finite permutation group G, subgroup of the symmetric group Sl, we determine the permutations involving only members of G as l-patterns, i.e. avoiding all patterns in the set Sl\G. The set of all n-permutations with this property constitutes again a permutation group. We consequently refine and strengthen the classification of sets of permutations closed under pattern involvement and composition that is due to Atkinson and Beals.

CEMAT - Center for Computational and Stochastic Mathematics