Small Representations of Permutation Groups by Embedding the Domain in an Ordered Set

TL;DR

This paper introduces a novel method to represent permutation groups as automorphism groups of ordered sets, often using fewer points than previous methods, especially for imprimitive groups.

Contribution

It provides a new representation technique for permutation groups via ordered sets, improving efficiency for many imprimitive groups.

Findings

Fewer points used in representations of many imprimitive groups.

Representation captures the permutation group's action on a subset.

Method generalizes existing representations with improved efficiency.

Abstract

We present a representation for permutation groups as the automorphism group of an ordered set $U$ such that the automorphism group's action on a subset $T\subseteq U$ is the permutation group itself. For many imprimitive permutation groups, the representation uses fewer points than the smallest representations to date.