CHAPTER 5

Translation Planes

Michael KALLAHER

Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164-3113, USA

Contents

Introduction and historical survey

1. Basic theory

2. Classes of translation planes

2.1. Generalized André systems and planes

2.2. André systems and planes

2.3. Nearfields and their planes

2.4. Semifields and their planes

2.5. Derived translation planes

2.6. Generalized Hall planes, Hall planes

2.7. Johnson planes

2.8. Ostrom--Foulser planes

2.9. Hering--Ott--Schaeffer planes

2.10. Likeable planes and Walker planes

2.11. Lüneburg planes

2.12. Kantor planes

2.13. Suetake planes. Hering plane of order 27

2.14. Bartolone--Ostrom planes

2.15. Flocks and translation planes. The Fisher planes

3. Dimension and collineations

4. Collineation groups

4.1. Affine perspectivities

4.2. Planar collineations

4.3. Transitive collineation groups

5. Conclusion: two important problems

References