Foundations of Translation Planes

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ISBN: 9780824706098 | 0824706099
Cover: Hardcover
Copyright: 7/13/2001

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An exploration of the construction and analysis of translation planes to spreads, partial spreads, co-ordinate structures, automorphisms, autotopisms, and collineation groups. It emphasizes the manipulation of incidence structures by various co-ordinate systems, including quasisets, spreads and matrix spreadsets. The volume showcases methods of structure theory as well as tools and techniques for the construction of new planes.

Preface

An Overview

(8)

Andre's Theory of Spreads

(16)

Introduction

(1)

Group Partitions

(5)

The Andre Characterization of Spreads

(10)

Andre's Spread Isomorphism Theorem

(3)

The Fundamental Theorem of Translation Planes

(2)

Spreads in PG (3, K)

(14)

Introduction

(3)

Conical Spreads

(2)

Derivation of Spreads in PG (3, q)

(2)

s-Inversion and s-Square

(7)

Partial Spreads and Translation Nets

(20)

Introduction

(2)

Isomorphisms of Partial Spreads

(2)

Translation Nets

(1)

Sub (Partial) Spreads; Planar Groups

(3)

Kernel Subplanes of a Spread

(3)

Intersections of Partial Spreads: Lattice Structure

(3)

Rational Partial Spreads and Foulser-Covers

(6)

Spreadsets and Partial Spreadsets

(10)

Introduction

(1)

Permutation Sets

(1)

Spreadsets and Kernels

(3)

Desarguesian and Matrix Spreadsets

(5)

Geometry of Spreadsets: ∇(τ)

(26)

Introduction

(1)

The Gradient ∇(τ) of a Partial Spreadset τ

(6)

Coordinatizing Finite Spreads

(3)

Basis Changes for Spreadsets

(2)

Induced Partial Spreadsets

(8)

Rational Desarguesian Partial Spreadsets

(6)

Coordinatization by Spreadsets: General Cases

(24)

Introduction

(2)

Fundamental Theorem for Spreadsets

(4)

Coordinatization: by Axes Identification

101

(5)

Induced Coordinatization

106

(2)

Central Collineations: Applications of Matrix Spreadsets

108

(11)

Partial Qualsifields

119

(16)

Introduction

119

(1)

Partial Quasifields: Subsystems and Rational Partial Spreads

119

(7)

Spreadset-Characterizations of Quasifields

126

(3)

Isomorphism Among (Partial) Quasifields

129

(6)

Coordinatization by (Partial) Quasifields

135

(20)

Introduction

135

(1)

Coordinatization of Partial Spreadsets

135

(6)

Spreads Coordinatized by Quasifields: Lattice of Subquasifields; Planar Groups

141

(6)

Label-Irrelevance; Frame-Transitivity; Planar Groups

147

(2)

Symmetric Autotopism Groups and Non-isomorphic Partial Quasifields

149

(6)

Rational Desarguesian Nets

155

(8)

Introduction

155

(1)

Desarguesian Nets from Partial Quasifields

155

(4)

Characterization of Rational Partial Spreads by Partial Quasifields

159

(4)

Quasigroups, Loops and Nuclei

163

(12)

Introduction

163

(1)

Quasigroups and Loops

164

(4)

Nuclei of Loops

168

(3)

Nuclei for Pseudorings

171

(4)

(Pre) Quasifields: Algebraic Axioms and Autotopisms

175

(40)

Introduction

175

(1)

Pseudoquasifields

176

(3)

The Geometry of Pseudoquasifields

179

(3)

Coordinatizing Spreadsets by Prequasifields

182

(5)

Isotopisms for Quasifields

187

(10)

Affine Central Collineations of II(Q)

197

(9)

Nearfields, Semifields and Desarguesian Planes

206

(2)

Spreadsets Containing Fields

208

(7)

The Kernel of Spreadsets and Quasifields

215

(14)

Introduction

215

(1)

The Schur-Kernel of a Spreadset

216

(4)

Kernel for Quasifields

220

(4)

Kernel of Prequasifields

224

(5)

Quadratics of Two-Dimensional Quasifields: Hall Systems

229

(14)

Introduction

229

(5)

Two-Dimensional Transitive Quasifields

234

(2)

Hall Quasifields

236

(7)

Spreads in Projective Spaces

243

(12)

Introduction

243

(3)

Reguli and Pseudo-Reguli

246

(6)

Reguli-Covered Spreads: Bruck-Bose Theorem

252

(3)

Digression on Andre's Homology Theorem

253

(2)

Kernel Subplanes across Desarguesian Nets

255

(8)

Introduction

255

(4)

Geometric Structure

259

(1)

The Center of a Finite Semifield Is a Geometric Invariant

260

(3)

Derivation of Finite Spreads

263

(12)

Introduction

263

(1)

Affine Plane Derivation

264

(2)

Derivation in Finite Spreads

266

(9)

Foulser's Covering Theorem

275

(8)

Introduction

275

(3)

The Proof of Foulser's Covering Theorem

278

(5)

Structure of Baer Groups

283

(14)

Introduction

283

(1)

Baer Automorphisms: Fundamental Matrix Identities

284

(6)

Generalized Hall Planes

290

(3)

Structure of Baer Groups

293

(4)

Frobenius Complements, p-Primitive Collineations, and Klein 4-Groups

297

(8)

Frobenius Groups on Translation Planes

297

(2)

p-Primitive Collineations

299

(2)

Klein 4-Groups on Odd-Order Spreads

301

(4)

Large Planar Groups

305

(12)

Introduction

305

(1)

Planar p-Groups

306

(4)

Tangentially Transitive Translation Planes

310

(4)

Generalized Tangentially Transitive Planes

314

(3)

Finite Generalized Andre Systems and Nearfields

317

(22)

Introduction

317

(1)

Construction of Generalized Andre Systems

317

(8)

Cyclic Groups in λ-Systems

325

(1)

Andre Systems

326

(2)

Highest Prime-Power Divisors of a - 1 Dividing ad - 1

328

(5)

Dickson Nearfields

333

(6)

Elation Net Theory

339

(22)

Algebra Generated by a Matrix A

339

(1)

Generalized Elations

340

(7)

Tensor Product Extensions of Partial Spreads

347

(4)

Properties of SL (2, q)

351

(2)

Ostrom's Theorem

353

(8)

Baer-Elation Theory

361

(22)

Introduction

361

(1)

Baer-Elation Theory: Odd Order Case

361

(10)

Incompatibility Theory: Even Order

371

(12)

Maximal Elation Groups and Baer Involutions

372

(4)

Large Baer Groups and Elations

376

(4)

Baer Groups with Non-Normalizing Elation Groups

380

(3)

Semifields

383

(18)

Introduction

383

(1)

General Remarks on Semifields

383

(1)

The Knuth Commutative Semifields

384

(4)

Twisted Fields

388

(6)

Vaughan Polynomials

389

(1)

The Vaughan Polynomial for P and Non-Commutativity of Twisted Fields

390

(4)

Generalized Twisted Fields

394

(4)

Some Two-Dimensional Semifields

398

(3)

Simple T-Extensions of Derivable Nets

401

(14)

Introduction

401

(1)

Spreadsets Containing Fields

402

(4)

T -Extensions

406

(1)

T -Extension/Transpose

407

(2)

T -Distortion

409

(3)

The `New' Multiplication

410

(2)

The Construction of 2-Dimensional Semifields

412

(3)

Cyclic Semifields

415

(6)

Introduction

415

(2)

T - Cyclic GL(2, q)-spreads

417

(4)

Baer Groups on Parabolic Spreads

421

(16)

Introduction

421

(1)

Derivable Linespreads

422

(7)

Parabolic Spreads in PG(3, q)

429

(8)

Lifting and Quasifibarations

437

(30)

Introduction

437

(2)

Quais-Spreadsets

439

(3)

Quasifibrations

442

(3)

Parabolic Linespreads: Infinite Case

445

(5)

Lifting Quasi-Spreadsets

450

(5)

(Infinite) Linespreads Also Life to Linespreads

455

(2)

Lifted Semifields

457

(5)

Semifields Lift to Semifields

458

(1)

Lifted Pappian Spreads

459

(3)

Lift-Dualize-Derive

462

(1)

Lifted Quasifibrations: Infinite Case

463

(1)

Chains of lIfts and Retractions

464

(3)

Mixed Tangentially Transitive Planes

467

(14)

Introduction

467

(1)

Transposed Spreads

468

(2)

Critical Deficiency for Translation Planes

470

(2)

Derive-Transpose = Transpose-Derive

472

(1)

Mixed Transitivity

473

(1)

Planes of Order 16

474

(5)

Parallelisms Associated with LR and JW

479

(2)

Maximal Partial Spreads

481

(8)

Introduction

481

(1)

The Ubiquity of Maximal Partial Spreads

482

(2)

Even Order

484

(5)

Foulser-Johnson SL(2, q)-Theorem

489

(10)

Introduction

489

(1)

The Examples

490

(3)

Desarguesian Planes

490

(1)

Hall Planes

490

(1)

Hering and Ott-Schaeffer Planes

490

(3)

The Three Walker Planes of Order 25

493

(1)

Spreads in PG(3, q) Admitting SL(2, q)

493

(1)

Arbitrary Dimension

494

(1)

Applications

495

(4)

Elations and Large Baer Groups

496

(3)

APPENDICES

499

(28)

Appendix I: Vector Spaces over Skewfields

499

(2)

Appendix II: Linear Representations

501

(2)

Appendix III: Affine Spaces

503

(3)

Appendix IV: Projective Spaces

506

(1)

Appendix V: Incidence Structures and Parallelisms

507

(5)

Appendix VI: Equivalence of Affine and Projective Spaces

512

(2)

Appendix VII: Affine and Projective Planes

514

(10)

Appendix VIII: Permutation Group Theory

524

(3)

Bibliography

527

(8)

Index

535

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Foundations of Translation Planes

Summary

Author Biography

Table of Contents

Supplemental Materials

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Foundations of Translation Planes

Summary

Author Biography

Table of Contents

Supplemental Materials

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