1st Edition

Combinatorics of Spreads and Parallelisms




ISBN 9781439819463
Published June 3, 2010 by CRC Press
674 Pages

USD $280.00

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Book Description

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.

Table of Contents

Partitions of Vector Spaces
Quasi-Subgeometry Partitions
Finite Focal-Spreads
Generalizing André Spreads
The Going Up Construction for Focal-Spreads

Subgeometry Partitions
Subgeometry and Quasi-Subgeometry Partitions
Subgeometries from Focal-Spreads
Extended André Subgeometries
Kantor’s Flag-Transitive Designs
Maximal Additive Partial Spreads

Subplane Covered Nets and Baer Groups
Partial Desarguesian t-Parallelisms
Direct Products of Affine Planes
Jha-Johnson SL(2, q) × C-Theorem
Baer Groups of Nets
Ubiquity of Subgeometry Partitions

Flocks and Related Geometries
Spreads Covered by Pseudo-Reguli
Flocks
Regulus-Inducing Homology Groups
Hyperbolic Fibrations and Partial Flocks
j-Planes and Monomial Flocks

Derivable Geometries
Flocks of α-Cones
Parallelisms of Quadric Sets
Sharply k-Transitive Sets
Transversals to Derivable Nets
Partially Flag-Transitive Affine Planes
Special Topics on Parallelisms

Constructions of Parallelisms
Regular Parallelisms
Beutelspacher’s Construction of Line Parallelisms
Johnson Partial Parallelisms

Parallelism-Inducing Groups
Parallelism-Inducing Groups for Pappian Spreads
Linear and Nearfield Parallelism-Inducing Groups
General Parallelism-Inducing Groups

Coset Switching
Finite E-Switching
Parallelisms over Ordered Fields
General Elation Switching
Dual Parallelisms

Transitivity
p-Primitive Parallelisms
Transitive t-Parallelisms
Transitive Deficiency One
Doubly Transitive Focal-Spreads

Appendices
Open Problems
Geometry Background
The Klein Quadric
Major Theorems of Finite Groups
The Diagram

Bibliography

Index

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Author(s)

Biography

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