Synchronising primitive groups of diagonal type exist

TL;DR

This paper presents the first known example of a synchronising permutation group of diagonal type, specifically demonstrating that certain groups involving PSL(2,q) are separating and synchronising for specific prime powers.

Contribution

It provides the first explicit example of a diagonal type synchronising group and analyzes its properties, expanding understanding of permutation group classifications.

Findings

PSL(2,13)×PSL(2,13) acting diagonally is separating and synchronising.

PSL(2,17)×PSL(2,17) acting diagonally is separating and synchronising.

Such groups are non-spreading for all prime powers q.

Abstract

Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that $\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$ acting in its diagonal action on $\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and $q=17$. Furthermore, we show that such groups are non-spreading for all prime powers $q$.