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