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

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 $rank(G)=3$.

13 pages