Block-transitive $3$-$(v,k,1)$ designs associated with alternating groups

TL;DR

This paper classifies certain symmetric combinatorial designs with automorphism groups related to alternating groups, showing uniqueness and specific group actions for the 3-(10,4,1) design.

Contribution

It proves that if an almost simple automorphism group with an alternating socle acts block-transitively on a 3-(v,k,1) design, then the design is uniquely the 3-(10,4,1) with specific automorphism groups.

Findings

The 3-(10,4,1) design is unique under these conditions.

Automorphism groups are limited to PGL(2,9), M10, or S6:Z2.

The group action is flag-transitive.

Abstract

Let $\mathcal{D}$ be a nontrivial $3$-$(v,k,1)$ design admitting a block-transitive group $G$ of automorphisms. A recent work of Gan and the second author asserts that $G$ is either affine or almost simple. In this paper, it is proved that if $G$ is almost simple with socle an alternating group, then $\mathcal{D}$ is the unique $3$-$(10,4,1)$ design, and $G=\mathrm{PGL}(2,9)$, $\mathrm{M}_{10}$ or $\mathrm{Aut}(\mathrm{A}_6 )=\mathrm{S}_6:\mathrm{Z}_2$, and $G$ is flag-transitive.