The non-existence of block-transitive subspace designs

Let \(q\) be a prime power and \(V\cong\mathbb{F}_q^d\). A \(t\)-\((d,k,\lambda)_q\) design, or simply subspace is pair \(\mathcal{D}=(V,\mathcal{B})\), where \(\mathcal{B}\) subset of the set all \(k\)-dimensional subspaces \(V\), with property that each \(t\)-dimensional \(V\) contained in precisely \(\lambda\) elements \(\mathcal{B}\). Subspace designs are \(q\)-analogues balanced incomplete block designs. Such design called block-transitive if its automorphism group \(\mathrm{Aut}(\mathcal{D})\) acts transitively on It shown here \(t\geq 2\) \(\mathcal{D}\) then trivial, is, \(V\).Mathematics Subject Classifications: 05E18, 05B99