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Divisible Semiplanes, Arcs, and Relative Difference Sets

Published online by Cambridge University Press:  20 November 2018

Dieter Jungnickel
Dieter Jungnickel*
Affiliation:
Justus-Liebig-Universität Giessen, Giessen, F.R. Germany
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In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following:

  • (i) S is the empty set.

  • (ii) S consists of a line L with all its points and a point p with all the lines through it.

  • (iii) S is a Baer subplane of Π.

We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 4 , 01 August 1987 , pp. 1001 - 1024
Copyright
Copyright © Canadian Mathematical Society 1987

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