Block-transitive automorphism groups on 2-designs with block size 4

Abstract

The main aim of this paper is to study 2-$(v,4,\lambda )$ designs admitting a block-transitive automorphism group $G$. We indeed determine all such designs when $G$ is point-imprimitive or point-primitive group of product type.

Introduction

Definition 1.1

For positive integers $t<k<v-1$ and $\lambda$, a $t$-$(v,k,\lambda )$ design $\mathcal{D}$ is an incidence structure $(\mathcal{P},\mathcal{B})$ satisfying the following properties:

• $\mathcal{P}$ is a set of $v$ elements, called points,

• $\mathcal{B}$ is a set of $b$ $k$-subsets of $\mathcal{P}$, called blocks,

• Every $t$-subset of $\mathcal{P}$ is contained in exactly $\lambda$ blocks.

Since all the blocks have the same size $k$, it follows that each point belongs to the same number $r$ of blocks.

An automorphism of $\mathcal{D}$ is a permutation of the points which also permutes the blocks. The set of all automorphisms of $\mathcal{D}$ with the composition of maps is a group, denoted by $\mathrm{Aut}\left(\mathcal{D}\right)$. Let $G\le \mathrm{Aut}\left(\mathcal{D}\right)$. If $G$ is a primitive permutation group on the point set $\mathcal{P}$, then $G$ is called point-primitive, otherwise point-imprimitive. A flag of $\mathcal{D}$ is a point–block pair $(\alpha ,B)$ with $\alpha \in B$, and $G$ is called flag-transitive if it acts transitively on the set of flags. Block-transitivity is defined similarly. A set of blocks of an incidence structure $\mathcal{D}$ is called a set of base blocks with respect to an automorphism group $G$ of $\mathcal{D}$ if it contains exactly one block from each $G$-orbit on the block set. In particular, if $G$ is a block-transitive automorphism group of $\mathcal{D}$, then any block $B$ is a base block of $\mathcal{D}$.

It follows from a result of Ray-Chaudhuri and Wilson [7] that a block-transitive automorphism group of a 4-design is 2-homogeneous, and hence primitive on points. Thus, for a block-transitive point-imprimitive $t$-design, the parameter $t$ is at most 3. It was shown in [2] that for a block-transitive and point-imprimitive $3$-$(v,k,\lambda )$ design $\mathcal{D}$, $v\le \left(\genfrac{}{}{0ex}{}{k}{2}\right)+1$ if $\mathrm{Aut}\left(\mathcal{D}\right)$ preserves a partition with either 2 parts or parts of size 2. It is an open question whether or not this better holds bound in general for 3-design. In 1987, Davies [4] proved that, for a given value of $\lambda$, the block size $k$ of a flag-transitive point-imprimitive $2$-design is bounded. In 1989, Delandtsheer and Doyen [5] proved that, if $\mathcal{D}$ is a block-transitive point-imprimitive $2$-$(v,k,\lambda )$ design, then $v\le {(\left(\genfrac{}{}{0ex}{}{k}{2}\right)-1)}^2$ (see Lemma 2.3). It follows immediately that, for a block-transitive point-imprimitive $2$-$(v,k,\lambda )$ design, the number $v$ of points cannot be too large relative to the block size $k$. Thus, if $\mathcal{D}$ is a $2$-$(v,k,\lambda )$ design with a block-transitive and point-imprimitive automorphism group $G$, then $k\ge 4$. The upper bound for 2-designs is attained has been almost completely answered in [2] when $k\ne 3,4,5,8$. The classification is an open problem for these values of $k$. In this paper, we give a complete classification of $2$-$(v,4,\lambda )$ designs admitting a block-transitive point-imprimitive automorphism group.

We now state the first result of this paper:

Theorem 1

Let $\mathcal{D}$ be a $2$-$(v,4,\lambda )$ design, and $G$ be a block-transitive point-imprimitive automorphism group of $\mathcal{D}$. Then $v=25$ and

$$\lambda \in \{2_2,4_4,5_3,6_1,8_7,10_{22},12_1,18_1,20_5,24_3,25_1,40_2,50_2,72_1,100_1,150_1\}.$$

Remark 1

The notation $\lambda =a_n$ means that $\lambda =a$ and there are exactly $n$ pairwise non-isomorphic such $2$-$(25,4,a)$ designs. The base block and automorphism group of each design are given in Table 2.

For the point-primitive case, most research on block-transitive $2$-$(v,k,\lambda )$ designs hitherto has concerned the affine and almost simple types. However, there are very few studies that fully explore the block-transitive point-primitive $2$-$(v,k,\lambda )$ designs of product type. Recently, Zhan et al. determine all flag-transitive point-primitive $2$-$(v,4,\lambda )$ designs of product type in [9]. Focusing on 2-$(v,4,\lambda )$ designs, we analyse the case in which the automorphism group is block-transitive point-primitive of product type and get the following:

Theorem 2

Let $\mathcal{D}=(\mathcal{P},\mathcal{B})$ be a $2$-$(v,4,\lambda )$ design with a block-transitive point-primitive automorphism group $G$. Assume that $G$ has a product action on the set $\mathcal{P}$. Then $\mathrm{Soc}\left(G\right)=A_5\times A_5$, and $\mathcal{D}$ is a unique $2$-$(25,4,12)$ design, a unique $2$-$(25,4,18)$ design or a unique $2$-$(25,4,72)$ design.