Permutation groups, pattern involvement, and Galois connections

TL;DR

This paper explores the relationship between permutation groups and pattern involvement, characterizing groups arising from pattern avoidance and automorphism groups through Galois connections.

Contribution

It introduces a novel characterization of permutation groups via pattern avoidance and automorphism groups using Galois connections.

Findings

Permutation groups arising from pattern involvement are characterized as automorphism groups.

A Galois connection between permutation groups and relations is established and analyzed.

Closed sets and kernels in the Galois connection are described as automorphism groups.

Abstract

There is a connection between permutation groups and permutation patterns: for any subgroup $G$ of the symmetric group $S_\ell$ and for any $n \geq \ell$, the set of $n$-permutations involving only members of $G$ as $\ell$-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.