Steiner Loops of Affine Type

Abstract

Steiner loops of affine type are associated to arbitrary Steiner triple systems. They behave to elementary abelian 3-groups as arbitrary Steiner Triple Systems behave to affine geometries over $${\mathrm {GF}}(3)$$ GF ( 3 ) . We investigate algebraic and geometric properties of these loops often in connection to configurations. Steiner loops of affine type, as extensions of normal subloops by factor loops, are studied. We prove that the multiplication group of every Steiner loop of affine type with n elements is contained in the alternating group $$A_n$$ A n and we give conditions for those loops having $$A_n$$ A n as their multiplication groups (and hence for the loops being simple).

A new family of additive designs

“…Moreover, it is shown that symmetric and affine resolvable 2-designs are additive and that, for these designs, and for a suitable choice of G, the blocks are exactly the (unordered) k-tuples of elements in P which sum up to zero, so that the automorphism group of D coincides with the stabilizer of P in the automorphism group of G. On the contrary, it is shown that the only additive Steiner triple systems are the point-line designs of AG(d, 3) and PG(d, 2) (cf. also [11] and [12]).…”

A visual representation of the Steiner triple systems of order 13

“…It is worth mentioning that algebraic representations of the STS(13)s were recently given in [31] and in [12,Proposition 4]. In the former case, an incidence structure isomorphic to a Steiner triple system of order 13 is constructed by defining a set B of twenty-six vectors in the 13-dimensional vector space V over the finite field F 5 , with the property that there exist precisely thirteen 6-subsets of B whose elements sum up to zero in V, whereas in the latter case the STS(13)s are described as sections of A(13)/A (12) in the alternating group A(13).…”