The Non-Existence of Block-Transitive Subspace Designs

Abstract

Let $q$ be a prime power and $V\cong {\mathbb{F}}_q^n$. A $t$-$(n,k,\lambda {)}_q$ design, or simply a subspace design, is a pair $\mathcal{D}=(V,\mathcal{B})$, where $\mathcal{B}$ is a subset of the set of all $k$-dimensional subspaces of $V$, with the property that each $t$-dimensional subspace of $V$ is contained in precisely $\lambda$ elements of $\mathcal{B}$. Subspace designs are the $q$-analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group $\mathrm{Aut}(\mathcal{D})$ acts transitively on $\mathcal{B}$. It is shown here that if $t\ge 2$ and $\mathcal{D}$ is a block-transitive $t$-$(n,k,\lambda {)}_q$ design then $\mathcal{D}$ is trivial, that is, $\mathcal{B}$ is the set of all $k$-dimensional subspaces of $V$.