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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.

Central automorphisms of Laguerre planes

Makowiecka H., Matraś A.

Demonstratio Mathematica

1998

Vol. 31, nr 4

753-763

Any automorphism of a Benz plane having at least one fixed point induces a collineation on the projective extension of the residual affine plane with reference to this point. When this collineation is a central automorphism , the initial automorphism is called the central automorphism (or central-axial automorphism, cf.([3]). In this paper we present an analytical description of central automorphisms of a miguelian Laguerre planes with the characteristic different from two. This description is applied to find transitive groups of homotheties and translations of types occuring in the classification theorems of R. Kleinewillinghofer ([2]). Some examples over an arbitrary commutative field are constructed, the other over the finite field Z3 and z5. It is interesting that two types of the Kleinewillinghofer classification ( [2] ) appear only as automorphism subgroups of finite plane of order three or five. This will give a clear characterization of these planes. Throughaut we assume that the characteristic of a plane is not equal to two.