Permutation groups arising from pattern involvement

TL;DR

This paper characterizes permutation groups formed by permutations that only involve members of a given subgroup as patterns, refining the classification of pattern-involved permutation sets.

Contribution

It provides a complete description of permutation groups arising from pattern involvement for any finite permutation group, strengthening previous classifications.

Findings

Permutation groups formed by pattern involvement are characterized.

The classification of pattern-involved permutation sets is refined.

Sets of permutations closed under pattern involvement form permutation groups.

Abstract

For an arbitrary finite permutation group $G$, subgroup of the symmetric group $S_\ell$, we determine the permutations involving only members of $G$ as $\ell$-patterns, i.e., avoiding all patterns in the set $S_\ell \setminus G$. 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.