Group Embeddings of (n,k) Configurations

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Ens, Eric
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An (n, k) configuration is a set of n “points” and n “lines” such that each point lies on k lines and each line contains k points. Motivated by the geometric definition of a group law on non-singular cubic curves, we define the concept of group embeddability of (n, k) configuration C as a mapping g of C into an abelian group G such that a set of k points {P1 , P2 , ..., Pk } are collinear in the configuration C if and only if ∑ g (Pi ) = 0 in the group G. Here we classify the set of all (n, 3) configurations for n ≤ 11 as well as some other notable configurations which can be embedded into abelian groups. Here we use the notation introduced by Branko Grünbaum [2]. The following theorems are proved in this thesis: n (n, 3) 7 Fano Plane 8 (8, 3) group Z2 × Z2 × Z2 Z3 × Z3 9 Of the three configurations, two are embeddable in groups. 10 Of the 10 configurations, five are embeddable in groups. 11 Of the 31 configurations, 9 have group embeddings. But for the first three examples (n = 7, 8 and the Pappus configuration), all other embeddability theorems proved here are new. In doing so we develop several different techniques for finding a group embedding or proving that no such embedding exists. Some ideas in this thesis were inspired by the late Professor N. S. Mendelsohn. For example, group embeddings can be thought of as extensions of configurations to Mendelsohn Triple Systems (see [8], [10]). In fact, configurations naturally give rise to partial quasigroups and adding the “missing triples” including the so-called "tangential relations" are the essential ideas behind the Mendelsohn triple Systems [8].
Mathematics, Configurations