The distinguishing number of quasiprimitive and semiprimitive groups

TL;DR

This paper investigates the distinguishing number of certain permutation groups, proving that most quasiprimitive and semiprimitive groups have a distinguishing number of two, with a specific exception.

Contribution

It extends previous work by showing that all imprimitive quasiprimitive groups and most non-quasiprimitive semiprimitive groups have a distinguishing number of two, except for one specific case.

Findings

All imprimitive quasiprimitive groups have distinguishing number two.

All non-quasiprimitive semiprimitive groups have distinguishing number two.

The group GL(2,3) acting on 8 vectors has distinguishing number three.

Abstract

The distinguishing number of $G \leqslant \sym(\Omega)$ is the smallest size of a partition of $\Omega$ such that only the identity of $G$ fixes all the parts of the partition. Extending earlier results of Cameron, Neumann, Saxl and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for $\mathrm{GL}(2, 3)$ acting on the eight non-zero vectors of $\mathbb F_2^3$, which has distinguishing number three.