The non-existence of block-transitive subspace designs

Let q be a prime power and V^∼= F^dq. A t-(d, k, λ)_q design, or simply a subspace design, is a pair D = (V, B), where 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 λ elements of B. Subspace designs are the q-analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group Aut(D) acts transitively on B. It is shown here that if t ≥ 2 and D is a block-transitive t-(d, k, λ)_q design then D is trivial, that is, B is the set of all k-dimensional subspaces of V.