Translation Planes

Frontmatter

Chapter I. Introduction

Abstract

In this chapter we shall collect the basic results about translation planes which will be used throughout the book. An exception is section 4 where we apply the general theory for the first time giving a characterization of pappian planes.

Heinz Lüneburg

Chapter II. Generalized André Planes

Abstract

We now turn to the investigation of a large class of translation planes which has been fairly well studied. Planes belonging to this class occur in several interesting instances.

Heinz Lüneburg

Chapter III. Rank-3-Planes

Abstract

This chapter starts with Wagner’s celebrated theorem that finite line transitive affine planes are translation planes. This theorem is used in the proof of Kallaher’s & Liebler’s theorem on affine planes of rank 3. Finally, rank-3-planes with an orbit of length 2 on l_∞ are investigated.

Heinz Lüneburg

Chapter IV. The Suzuki Groups and Their Geometries

Abstract

In this chapter we investigate the Suzuki groups as well as the Möbius planes and translation planes belonging to them. The investigations culminate in R. Liebler’s characterization of the Lüneburg planes. This chapter is a blending of my set of lecture notes 1965b and an unpublished set of lecture notes by A. Cronheim. I would like to thank him very much indeed for allowing me to incorporate his notes into this chapter.

Heinz Lüneburg

Chapter V. Planes Admitting Many Shears

Abstract

Next we collect results about unitary groups, we prove a characterization of A₅ which I extracted from J. Assion’s Diplomarbeit, and we give a characterization of Galoisfields of odd characteristic due to A. A. Albert. All this is done in order to prove Hering’s & Ostrom’s theorem on collineation groups of translation planes generated by shears.

Heinz Lüneburg

Chapter VI. Flag Transitive Planes

Abstract

In this chapter we give Huppert’s description of all finite soluble 2-transitive permutation groups and Foulser’s description of all soluble flag transitive collineation groups of finite affine planes. Using these and some characterizations of finite desarguesian projective planes involving the groups SL(2,q) and PSL(2,q), we are able to prove the theorem of Schulz and Czerwinski on finite translation planes admitting a collineation group acting 2-transitively on l_∞.

Heinz Lüneburg

Chapter VII. Translation Planes of Order q2 Admitting SL(2,q) as a Collineation Group

Abstract

This chapter gives the complete description of all translation planes of order q², having GF(q) contained in their kernels, and admitting SL(2,q) as a collineation group. Before we can give this description, we have to prove some results on ovals in finite desarguesian planes of odd order, among them Segre’s famous result that any oval in such a plane is a conic. Moreover, we have to investigate twisted cubics in projective 3-space and similar configurations in projective 3-spaces of characteristic 2.

Heinz Lüneburg

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