Department of Mathematics
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Ever since the astounding success of the cartesian geometry, algebraic tools have become the basis for many parts of mathematics and the topic of finite planes is no of exception. In this talk, we present one such algebraic concept, the so-called Netto Systems of Peter J. Cameron (page 116, [1]). Cameron uses cyclic groups, finite fields and Galois theory to represent a special class of Steiner triple systems as abelian groups in such a way that the sum of all the elements in a block equals zero in the group. In collaboration with N.S. Mendeloshn and Barry Wolk [2], I generalized this to Steiner n-tuples and applied this to show that the 10-point Desargues configuration is a Netto system of rank 4. By contrast, it is classically known that the familiar 9-point Pappus configuration is a Netto system of rank 2. Subsequently we proved (in [3]) that a large class of finite planes including the Desarguesian planes can be represented as generalized Netto systems. Here we present some of these examples and proofs. We also mention some open problems.

[1] Peter J. Cameron; Combinatorics: Topics, Techniques and Algorithms, Cambridge University Press, 1994
[2] N.S. Mendelsohn, R. Padmanabhan and Barry Wolk; Placement of Desargues Configuration, Geometriae Dedicata 40 (1991) 165-170.
[3] N.S. Mendelsohn, R. Padmanabhan and Barry Wolk, Group Representations of Finite Planes, Coxeter Legacy: a Conference at the University of Toronto, May 12-16, 2004.