A Theorem of Siemons and Wagner

TL;DR

This paper explores the properties of primitive permutation groups related to orbit lengths on subsets, extending previous classifications from 2-subsets to 3-subsets and identifying all 3-homogeneous groups with this property.

Contribution

It extends the classification of primitive groups with specific orbit length properties from 2-subsets to 3-subsets and lists all 3-homogeneous groups satisfying these conditions.

Findings

Classified primitive groups with the property for 3-subsets.

Listed all 3-homogeneous groups satisfying the property.

Extended previous results from 2-subsets to 3-subsets.

Abstract

In 1988 Siemons and Wagner describe a relationship between the lengths of $G$-orbits on subsets of a $G$-set $\Omega$. They highlighted the situation where $\Delta \subset \Omega$ with $|\Delta|=k,$ and $|\Delta^G|>|\Sigma^G|$ for all $(k+1)$-subsets, $\Sigma$ of $\Omega$ where $\Delta\subset\Sigma$. They went on to classify all primitive groups with this property for $k=2$. Here we address some questions about primitive permutation groups satisfying this property when $k=3$ and list all $3$-homogeneous groups satisfying this condition.