Combinatorics of Spreads and Parallelisms

Norman Johnson

June 3, 2010 by CRC Press
Reference - 674 Pages
ISBN 9781439819463 - CAT# K11024
Series: Chapman & Hall Pure and Applied Mathematics


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  • Covers general partitions of vector spaces, emphasizing focal-spreads and extended generalized André spreads
  • Explains how novel constructions provide retraction groups that allow a large number of new subgeometry partitions of projective spaces
  • Offers a thorough treatment of parallelisms of projective spaces
  • Lists more than 70 open problems that encompass many new areas of research


Combinatorics of Spreads and Parallelisms covers all known finite and infinite parallelisms as well as the planes comprising them. It also presents a complete analysis of general spreads and partitions of vector spaces that provide groups enabling the construction of subgeometry partitions of projective spaces.

The book describes general partitions of finite and infinite vector spaces, including Sperner spaces, focal-spreads, and their associated geometries. Since retraction groups provide quasi-subgeometry and subgeometry partitions of projective spaces, the author thoroughly discusses subgeometry partitions and their construction methods. He also features focal-spreads as partitions of vector spaces by subspaces. In addition to presenting many new examples of finite and infinite parallelisms, the book shows that doubly transitive or transitive t-parallelisms cannot exist unless the parallelism is a line parallelism.

Along with the author’s other three books (Subplane Covered Nets, Foundations of Translation Planes, Handbook of Finite Translation Planes), this text forms a solid, comprehensive account of the complete theory of the geometries that are connected with translation planes in intricate ways. It explores how to construct interesting parallelisms and how general spreads of vector spaces are used to study and construct subgeometry partitions of projective spaces.