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). 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 and we give conditions for those loops having A_n as their multiplication groups (and hence for the loops being simple).
Highlights
A quasigroup is a set L of elements endowed with a binary operation (◦)which does not need to be associative, and which is such that the equations a ◦ x = b and y ◦ a = b determine unique solutions x = a\b and y = b/a, its multiplication table is a latin square, and a loop is a quasigroup which has a neutral element Ω.As pointed out first by R
We give the definition of a Steiner loop of affine type and we prove the basic results in Sects. 1 and 2
In order to make the paper self-contained for a broader audience, in the Introduction we give the basic definitions of Steiner triple systems and of loops, and the role played by the group of left multiplications
Summary
Math which in particular is the following:. In the previous table, the reader can find an example to what we stated in Remark 11: since the set. (note that the latin square chosen here does not correspond to a group, nor a loop, because it is not symmetric). Each of the 9 entries of this latin square will correspond to a triple of SL , for instance {(z, −1), (ȳ, −1), (−x, −1)}. Φz,−xand Φȳ,−x , according to Definition 4 (vi), that is, we have to re-write the 9 triples given in (5), in the following two latin squares: the reader can freely choose a latin square on N = {−1, 0, 1} for. Once we have chosen the four latin squares Φz,ȳ , Φz,−ȳ , Φȳ,−z , Φ−ȳ,−z , we obtain the remaining 36 of the 70 triples STS(21). The remaining latin squares Φ-s are already determined by Definition 4 (ii), that is, by the fact that the addition table of the loop L is symmetric
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