This thesis deals primarily with different types of partitions of projective spaces, namely spreads and mixed partitions. It is well known that there is a close relationship between spreads of odd dimensional projective spaces and finite affine translation planes. Hence, more knowledge of spreads will hopefully one day lead to a better understanding of finite affine translation planes.

We discuss spreads constructed via a method described in General Galois Geometries by Hirschfeld and Thas. This method is carefully explained in Chapter 2 using geometric techniques to better understand the algebraic construction given in the book. The basic method starts with a mixed partition consisting of linear spaces together with Baer subspaces, and then lifts these spaces to a spread in a higher dimensional space. It is possible to construct the same translation plane directly from the associated mixed partition, although the construction of a translation plane from the associated spread is better known.

Chapter 3 provides the reader with some classical examples of mixed partitions. These partitions are constructed, via group theoretic techniques and are shown to generate the Desarguesian affine plane. Automorphism groups are discussed in Chapter 4. This work is used to determine properties of the collineation groups of such “geometrically lifted” spreads. We discuss the algebraic kernel of the affine planes arising from these spreads, and we prove some general properties about automorphisms of mixed partitions and their associated spreads.

In Chapter 5 we will generalize the lifting method given by Hirschfeld and Thas. From this, we prove a result about (n − 1)-spreads of $PG$(2n − 1, q) generating affine translation planes which are not n-dimensional over their kernel. This will lead to a general theory about distinct spreads lying in projective spaces of different dimension which generate .isomorphic translation planes. This result, together with a result of Lüneburg, provides a complete unifying theory for spreads which generate isomorphic translation planes.

Finally, Chapter 6 will provide the reader with some concrete examples of mixed partitions showing that this method is indeed useful for finding spreads. The author gives concrete constructions of several infinite families, and discusses the translation planes they construct via the Bose/Andrè model.