Almost simple groups with socle $L_n(q)$ acting on Steiner quadruple systems

TL;DR

This paper classifies Steiner quadruple systems admitting certain projective linear groups and shows that for higher dimensions, such groups cannot act as automorphisms on these systems.

Contribution

It provides a complete classification of Steiner quadruple systems with automorphism groups containing projective linear simple groups, highlighting the non-existence for n>2.

Findings

No Steiner quadruple systems admit such groups for n>2.

Complete classification of systems for n=2.

Groups act only in trivial cases for n>2.

Abstract

Let $N=L_n(q)$, {$n \geq 2$}, $q$ a prime power, be a projective linear simple group. We classify all Steiner quadruple systems admitting a group $G$ with $N \leq G \leq \Aut(N)$. In particular, we show that $G$ cannot act as a group of automorphisms on any Steiner quadruple system for $n>2$.