### Abstract

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

Original language | English |
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Pages (from-to) | 271-294 |

Number of pages | 24 |

Journal | Journal of Algebraic Combinatorics |

Volume | Volume 45, Issue 1 |

Publication status | Published - 2017 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

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## Cite this

Falcone, G., Caggegi, A., & Pavone, M. (2017). On the additivity of block designs.

*Journal of Algebraic Combinatorics*,*Volume 45, Issue 1*, 271-294.