6533b856fe1ef96bd12b311f

RESEARCH PRODUCT

On the additivity of block designs

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We show that symmetric block designs $${\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})$$D=(P,B) can be embedded in a suitable commutative group $${\mathfrak {G}}_{\mathcal {D}}$$GD 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 $${\mathfrak {G}}_\mathcal {D}$$GD that leave $$\mathcal {P}$$P invariant. In some special cases, the group $${\mathfrak {G}}_\mathcal {D}$$GD 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,ź) symmetric design of prime order p not dividing k, then $${\mathfrak {G}}_\mathcal {D}$$GD is (essentially) isomorphic to $$({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}$$(Z/pZ)v-12, 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/pZ)v-12.