We show that symmetric block designs D = (P, B) can be embedded in a suitable commutative group 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 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 GD that leave P invariant.In some special cases, the group GD can be determined uniquely by the parameters of D. For instance, if D is a 2 − (v, k, λ) symmetric design of prime order p not dividing k, then GD 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 .

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We classify all (finitely dimensional nilpotent Lie k-algebras h with 2-dimensional commutator ideals h', extending a known result to the case where h' is non-central and k is an arbitrary field. It turns\ud out that, while the structure of h depends on the field k if h' is central, it is independent of k if h' is non-central and is uniquely determined by the dimension of h. In the case where k is algebraically or real closed, we also list all nilpotent Lie k-algebras h with 2-dimensional central commutator ideals h' and dimk h < 12

We show that any affine block design D = (P, B) is a subset of a suitable commutative group G D , with the property that a k-subset of P is a block of D if and only if its k elements sum up to zero. As a consequence, the group of automorphisms of any affine design D is the group of automorphisms of G D that leave P invariant. Whenever k is a prime p, G D is an elementary abelian p-group. Keywords Affine block designs• Hadamard designs • Additive designs • Mathieu group M 11