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}.
Lingua originale | English |
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pagine (da-a) | 271-294 |
Numero di pagine | 24 |
Rivista | Journal of Algebraic Combinatorics |
Volume | 45 |
Stato di pubblicazione | Published - 2017 |
All Science Journal Classification (ASJC) codes
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- ???subjectarea.asjc.2600.2607???