Steiner Loops of Affine Type
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Steiner Loops of Affine Type ´ Giovanni Falcone, Agota Figula , and Carolin Hannusch Zur Erinnerung an Prof. Dr. Peter Plaumann (14.9.1937-14.4.2020)
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 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 An and we give conditions for those loops having An as their multiplication groups (and hence for the loops being simple). Mathematics Subject Classification. 05B07, 20N05, 51E10. Keywords. Steiner triple systems, steiner loops of affine type, multiplication groups of loops, finite geometries, commutative Moufang loop.
In this paper we want to describe Steiner triple systems by means of a standard tool in algebra, that is, the reduction to simple objects via a composition series, the corresponding extension theory, and the (possible) classification of these simple objects. Although the literature on Steiner triple systems is huge and authoritative, and the idea of associating a loop to a Steiner triple system can often be found there, this approach has not yet been examined in depth. We succeeded in the first two aims. As for a possible classification of simple Steiner triple systems, the situation is more delicate, first of all because the multiplication group of a simple loop is, in general, only a primitive group, The first author was supported by Universit` a di Palermo FFR. The second author was supported by the National Research, Development and Innovation Office (NKFIH) Grant No. K132951. The second and the third author were supported by the constructions EFOP3.6.1-16-2016-00022 and EFOP-3.6.3-VEKOP-16-2017-00002. These projects were supported by the European Union, co-financed by the European Social Fund. 0123456789().: V,-vol
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secondly because our conjectured question, whether a simple Steiner loop of affine type with n elements is a section of the alternating group on n letters over the stabilizer of one letter, remains open, even in the case where its multiplication group is simple. However, to the best of our knowledge, this has always been the case. The paper is organized as follows. 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 also the basic definitions of Steiner triple systems and of loops, and the role played by the group of left (or right) multiplications. Before switching over to the reduction to simple Steiner loops of affine type, we investigate in Sect. 3, which turns therefore to be more technical, the connection betwee
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