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Results in Mathematics

, 74:196 | Cite as

Embeddings and Ambient Automorphisms of the Pappus Configuration

  • Norbert Knarr
  • Markus J. StroppelEmail author
Article
  • 37 Downloads

Abstract

We classify embeddings (i.e., “labeled drawings”) of the Pappus configuration in projective planes over commutative fields, up to projective equivalence. Using pairs of field elements, we parameterize the space of classes of projectively equivalent embeddings, and then explicitly determine the group of ambient automorphisms (or dualities) for any given parameter pair, i.e., the subgroup of the group of all automorphisms (and dualities) of the abstract configuration that are induced by projective collineations (or dualities) leaving invariant the image under any embedding in the given class. It turns out that the existence of an ambient duality implies an ambient polarity. We show that these parameter pairs can be interpreted as pairs of cross ratios associated in a rather natural way with the embedded configuration. The number of equivalence classes of embeddings in a projective plane over a given finite field is determined. The groups that occur as full ambient groups are identified in the subgroup lattice of the full automorphism group of the abstract configuration. Finally, we use our results to understand embeddings of the Möbius–Kantor configuration.

Keywords

Pappus configuration Möbius–Kantor configuration automorphism embedding duality polarity 

Mathematics Subject Classification

Primary 51A10 Secondary 05B30 51E30 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.LExMath, Fakultät für Mathematik und PhysikUniversität StuttgartStuttgartGermany

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