Combinatorics of Spreads and Parallelisms: 1st Edition (Hardback) book cover

Combinatorics of Spreads and Parallelisms

1st Edition

By Norman Johnson

CRC Press

674 pages

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pub: 2010-06-03
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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


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


p-Primitive Parallelisms

Transitive t-Parallelisms

Transitive Deficiency One

Doubly Transitive Focal-Spreads


Open Problems

Geometry Background

The Klein Quadric

Major Theorems of Finite Groups

The Diagram



About the Author

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

About the Series

Chapman & Hall Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Algebra / General
MATHEMATICS / Geometry / General
MATHEMATICS / Combinatorics