Permutations on the random permutation

TL;DR

This paper classifies all symmetries and first-order definable structures within the random permutation by identifying exactly 39 closed supergroups of its automorphism group, using Ramsey-theoretic methods.

Contribution

It provides a complete classification of symmetries of the random permutation, answering a longstanding open problem with a novel Ramsey-theoretic approach.

Findings

39 closed supergroups of the automorphism group identified

All structures first-order definable in the random permutation classified

Exposes the full symmetry group structure of the random permutation

Abstract

The random permutation is the Fra\"iss\'e limit of the class of finite structures with two linear orders. Answering a problem stated by Peter Cameron in 2002, we use a recent Ramsey-theoretic technique to show that there exist precisely 39 closed supergroups of the automorphism group of the random permutation, and thereby expose all symmetries of this structure. Equivalently, we classify all structures which have a first-order definition in the random permutation.