Additivity of affine designs

We show that any affine block design D = (P, B) is a subset of a suitable commutative group G_D, with the property that a k-subset of P is a block of D if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design D is the group of automorphisms of G_D that leave P invariant. Whenever k is a prime p, G_D is an elementary abelian p-group.