Block-transitive automorphism groups on 3-designs with small block size

TL;DR

This paper classifies the structure of block-transitive automorphism groups of small block size 3-designs, showing they are either affine or almost simple if point-primitive, and describing specific cases if point-imprimitive.

Contribution

It provides a classification of automorphism groups for 3-designs with block size up to 6, distinguishing between primitive and imprimitive cases and detailing their structures.

Findings

Point-primitive groups are of affine or almost simple type.

Point-imprimitive groups correspond to a specific 3-(16,6,λ) design with known λ values.

The rank of such groups is always 3.

Abstract

The paper is an investigation of the structure of block-transitive automorphism groups of a 3-design with small block size. Let $G$ be a block-transitive automorphism group of a nontrivial $3$-$(v,k,\lambda)$ design $\mathcal{D}$ with $k\le 6$. We prove that if $G$ is point-primitive then $G$ is of affine or almost simple type. If $G$ is point-imprimitive then $\mathcal{D}$ is a $3$-$(16,6,\lambda)$ design with $\lambda\in\{4, 12, 16, 24, 28, 48, 56, 64, 84, 96, 112, 140\}$, and $rank(G)=3$.