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On some operations on finite affine planes applied to the theory of regular configurations

Makowiecka H.

Demonstratio Mathematica

2010

Vol. 43, nr 1

167-185

We consider some operations on affine planes which resemble the construction of a derived affine plane at a point of the Benz plane. We call them Benz-contractions (B-contractions) , distinguishing between chain contractions and generator contractions. We prove that the Pappos-Pascal configuration is the B-contraction of the affine plane of order 4 and we relate it to the Havlicek-Tietze configuration. We present a new (HT)o-configuration and research some problems of embeddability for (P-P), (H-T), and (HT)o. We propose a method of finding (n - 2) regular configurations on an arbitrary affine plane of order n. Among them are pairs of configurations with dual type and each such a pair can be completed with one point and n -f 1 lines to the initial plane. We prove that for an arbitrary n odd, the non-existence of the symmetric configuration ( n2-1/2, n+1/2) the non-existence of the projective plane of order n. On the basis of Gropp's article flOj, we solve some current problems concerning the existence of non-symmetric configurations with a natural index.