The non-existence of block-transitive subspace designs

Abstract

Let \(q\) be a prime power and \(V\cong\mathbb{F}_q^d\). A \(t\)-\((d,k,\lambda)_q\) 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 \(t\geq 2\) and \(\mathcal{D}\) is a block-transitive \(t\)-\((d,k,\lambda)_q\) design then \(\mathcal{D}\) is trivial, that is, \(\mathcal{B}\) is the set of all \(k\)-dimensional subspaces of \(V\).Mathematics Subject Classifications: 05E18, 05B99