Distinguishing Primitive Permutation Groups

TL;DR

This paper proves that any primitive permutation group acting on a set of size at least 336 has a distinguishing number of two, meaning it can be distinguished by a partition into two cells.

Contribution

The paper establishes a universal bound for the distinguishing number of primitive permutation groups with large degree, specifically proving it is two for degrees at least 336.

Findings

Primitive permutation groups of degree ≥ 336 have distinguishing number two.

The result applies to all primitive groups with sufficiently large degree.

Provides a bound that simplifies understanding symmetry-breaking in permutation groups.

Abstract

Let $G$ be a permutation group acting on a set $V$. A partition $\pi$ of $V$ is distinguishing if the only element of $G$ that fixes each cell of $\pi$ is the identity. The distinguishing number of $G$ is the minimum number of cells in a distinguishing partition. We prove that if $G$ is a primitive permutation group and $|V|\ge336$, its distinguishing number is two.