On the additivity of block designs

We show that symmetric block designs ({mathcal {D}}=({mathcal {P}},{mathcal {B}})) can be embedded in a suitable commutative group ({mathfrak {G}}_{mathcal {D}}) in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of ({mathrm {PG}}(d,2)) and ({mathrm {AG}}(d,3)). In both cases, the blocks can be characterized as the only k-subsets of (mathcal {P}) whose elements sum to zero. It follows that the group of automorphisms of any such design (mathcal {D}) is the group of automorphisms of ({mathfrak {G}}_mathcal {D}) that leave (mathcal {P}) invariant. In some special cases, the group ({mathfrak {G}}_mathcal {D}) can be determined uniquely by the parameters of (mathcal {D}). For instance, if (mathcal {D}) is a 2-((v,k,lambda )) symmetric design of prime order p not dividing k, then ({mathfrak {G}}_mathcal {D}) is (essentially) isomorphic to (({mathbb {Z}}/p{mathbb {Z}})^{frac{v-1}{2}}), and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of (mathcal {B}) can be characterized also as the v intersections of (mathcal {P}) with v suitable hyperplanes of (({mathbb {Z}}/p{mathbb {Z}})^{frac{v-1}{2}}).