Combinatorics of Spreads and Parallelisms

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.

Partitions of Vector Spaces Quasi-Subgeometry Partitions Finite Focal-SpreadsGeneralizing André SpreadsThe Going Up Construction for Focal-Spreads Subgeometry Partitions Subgeometry and Quasi-Subgeometry Partitions Subgeometries from Focal-SpreadsExtended André SubgeometriesKantor’s Flag-Transitive DesignsMaximal Additive Partial Spreads Subplane Covered Nets and Baer Groups Partial Desarguesian t-Parallelisms Direct Products of Affine PlanesJha-Johnson SL(2, q) × C-TheoremBaer Groups of NetsUbiquity of Subgeometry Partitions Flocks and Related GeometriesSpreads Covered by Pseudo-ReguliFlocksRegulus-Inducing Homology GroupsHyperbolic Fibrations and Partial Flocks j-Planes and Monomial Flocks Derivable Geometries Flocks of a-ConesParallelisms of Quadric SetsSharply k-Transitive SetsTransversals to Derivable NetsPartially Flag-Transitive Affine PlanesSpecial Topics on Parallelisms Constructions of Parallelisms Regular ParallelismsBeutelspacher’s Construction of Line Parallelisms Johnson Partial Parallelisms Parallelism-Inducing Groups Parallelism-Inducing Groups for Pappian Spreads Linear and Nearfield Parallelism-Inducing GroupsGeneral Parallelism-Inducing Groups Coset Switching Finite E-SwitchingParallelisms over Ordered Fields General Elation Switching Dual Parallelisms Transitivity p-Primitive ParallelismsTransitive t-ParallelismsTransitive Deficiency OneDoubly Transitive Focal-Spreads AppendicesOpen Problems Geometry Background The Klein QuadricMajor Theorems of Finite GroupsThe Diagram Bibliography Index

Norman L. Johnson is a professor in the Department of Mathematics at the University of Iowa.