Combinatorics of Spreads and Parallelisms

Contents......Page 8
Preface......Page 14
Part 1: Partitions of Vector Spaces......Page 21
0.1. Partitions of Vector Spaces......Page 23
0.3. Double-Spreads Constructed from t-Spreads......Page 27
0.4. Main Theorem on Finite Translation Geometries......Page 29
1. Collineation Groups of Translation Geometries......Page 30
1.1. Collineation Groups of Translation Planes......Page 32
2: Finite Focal-Spreads......Page 34
1. Towers of Focal-Spreads......Page 35
2. Focal-Spreads and Coordinatization......Page 37
2.1. Matrix Representation of Focal-Spreads......Page 38
3. k-Cuts and Inherited Groups......Page 39
3.1. Matrix Representation of k-Cuts......Page 42
3.2. Nearfield Focal-Extension Theorem......Page 44
5. The Dual Semifield Plane......Page 45
6. The Six Associated Semifields......Page 48
6.1. The ‘Transfer.’......Page 50
7. Symplectic Spreads......Page 51
8. Additive Focal-Spreads......Page 54
9.1. Design Lemma......Page 57
10.1. Focal-Spread Reconstruction Theorem......Page 58
10.2. Parallelisms of Designs......Page 60
3: Generalizing André Spreads......Page 61
1. A Primer on André Planes......Page 62
2. r-(sn, q)-Spreads......Page 63
2.1. Main Theorem on Replacements......Page 67
2.2. Extended André Replacements......Page 71
3. Multiple Extended Replacements......Page 73
4.1. Extended André Large Group Theorem......Page 75
0.2. Sperner Spread Construction Theorem......Page 78
1. The ‘Going Up’ Construction......Page 80
2. Generalization to Partitions with Many Spread Types......Page 82
2.1. sk-Generalization......Page 83
3. Going Up–Direct Sums......Page 86
Part 2: Subgeometry Partitions......Page 88
5: Subgeometry and Quasi-subgeometry Partitions......Page 90
0.1. When Is a Quasi-Subgeometry a Subgeometry?......Page 94
1. k-Cuts of Subgeometry Partitions......Page 99
2. Additive k-Cuts......Page 101
3. Right/Left Focal-Spreads......Page 102
4. Hyperplane Constructions......Page 104
4.1. Using Desarguesian Planes......Page 105
7: Extended André Subgeometries......Page 107
0.2. The Ebert-Mellinger r-(rn,q)-Spreads......Page 111
1. Kantor’s Class I......Page 113
2. Kantor’s Class II......Page 115
4. Subgeometries from Kantor’s Class I......Page 117
5. Subgeometries from Kantor’s Class II......Page 120
5.1. Additional Constructions......Page 123
5.2. Field Subgeometry Partitions......Page 124
9: Maximal Additive Partial Spreads......Page 126
1. Direct Sums of Semifields......Page 129
2. Subfields of Order 4 in Knuth Semifields......Page 132
3. The Commutative Kantor Semifields......Page 135
Part 3: Subplane Covered Nets and Baer Groups......Page 139
1. A Primer on Subplane Covered Nets......Page 141
1.1. Classification of Subplane Covered Nets......Page 143
2. Projective Spreads and Affine Nets......Page 144
2.1. A Bound on Partial Desarguesian t-Parallelisms......Page 151
2.2. Generalization to Arbitrary Skewfields......Page 152
11: Direct Products of Affine Planes......Page 154
0.3. n-Fold Products......Page 156
1. Desarguesian Products......Page 157
2. Left Spreads and Left Partial Spreads......Page 160
2.1. 2-Fold Products......Page 162
3. Desarguesian Left Parallelisms–Right Spreads......Page 163
3.1. Deficiency......Page 164
3.2. Deficiency One Partial Parallelisms......Page 165
12: Jha-Johnson SL(2, q)x C-Theorem......Page 168
13: Baer Groups of Nets......Page 176
1. Left and Right in Direct Product Nets......Page 178
1.1. When K is Ko......Page 182
2.1. Three Point-Baer Subplanes......Page 183
2.2. Nets with Two Point-Baer Subplanes......Page 187
2.3. Decomposable by a Point-Baer Subplane......Page 189
3. Translation Planes with Baer Groups......Page 191
4. Spreads in PG(3, K)......Page 200
14: Ubiquity of Subgeometry Partitions......Page 207
1. Jha-Johnson Lifting Theorem......Page 208
1.1. Hiramine-Matsumoto-Oyama Algebraic Lifting......Page 209
1.2. Johnson’s Retraction Theorem......Page 210
2. Double-Baer Groups......Page 213
3. Fusion of Baer Groups......Page 215
4. Double-Homology Groups......Page 218
5. Dempwolff’s Double-Baer Groups......Page 220
6. Subgeometry Partitions from Going Up......Page 222
7. Algebraic Lifting of Focal-Spreads......Page 223
Part 4: Flocks and Related Geometries......Page 227
1. Pseudo-Reguli......Page 229
2. Conical and Ruled Spreads over Skewfields......Page 235
1. Conical Flocks......Page 242
2. Hyperbolic Flocks......Page 246
4. Point-Baer Groups and Partial Flocks......Page 251
5. Deficiency One Partial Conical Flocks......Page 253
6. Deficiency One Partial Hyperbolic Flocks......Page 257
7.1. Payne-Thas Extension Theorem......Page 261
7.2. Johnson, Payne-Thas Theorem......Page 262
17: Regulus-Inducing Homology Groups......Page 263
0.3. Baker, Ebert, Penttila Hyperbolic Fibration Theo-rem......Page 264
0.4. Johnson’s Homology Theorem......Page 267
18: Hyperbolic Fibrations and Partial Flocks......Page 277
0.5. The Correspondence Theorem......Page 279
19: j-Planes and Monomial Flocks......Page 282
1. Hyperbolic Fibrations over the Reals......Page 286
2. Classification of the Real j-Planes......Page 290
2.1. Extension of Partial det K+-Partial Flocks......Page 292
Part 5: Derivable Geometries......Page 294
20: Flocks of alpha-Cones......Page 296
0.2. Elation Groups and Flokki Planes......Page 298
1. Maximal Partial Flokki and alpha-Flocks......Page 299
2.1. Baer Groups on alpha-Partial Flokki......Page 302
3. K-Flokki and Algebraic Lifting......Page 303
4. Net Replacement in the Hughes-Kleinfeld Planes......Page 304
4.1. The Group Permuting the alpha-Derivable Nets......Page 308
21: Parallelisms of Quadric Sets......Page 310
1. The Thas, Bader-Lunardon Theorem......Page 311
2. Parallelisms of Hyperbolic Quadrics......Page 312
3. Bol Planes......Page 314
3.1. The Hyperbolic Flocks of Riesinger......Page 317
4. Parallelisms of alpha-Cones......Page 318
22: Sharply k-Transitive Sets......Page 321
1. Subsets of PgammaL(n, K)......Page 322
2. Subsets in PgammaL(n, q) of Deficiency 1......Page 325
3. The Parallelisms of Bonisoli......Page 330
23: Transversals to Derivable Nets......Page 334
1. Algebraic Extensions of Derivable Nets......Page 335
2.1. Knarr’s Theorem on Geometric Extension......Page 337
3. Planar Transversal Extensions......Page 339
3.1. Planar Transversal-Spreads and Dual Spreads......Page 343
4. Semifield Extension-Nets......Page 344
24: Partially Flag-Transitive Affine Planes......Page 352
1. Derivable Affine Planes with Nice Groups......Page 357
1.1. Classification......Page 367
1.2. Groups of Order q6......Page 368
2. Skew Parallelisms......Page 371
Part 6: Constructions of Parallelisms......Page 375
1. Infinite Regular Parallelisms......Page 377
1.1. Multiple Derivation......Page 378
2.1. Betten and Riesinger 2-Secant Theorem......Page 380
3. Isomorphisms of Transitive Parallelisms......Page 383
3.2. The SL(2, K)-Partial Spreads......Page 384
4. Finite Regular Parallelisms......Page 385
5. The Penttila-Williams Construction......Page 386
5.1. The New SL(2, q) x C1+q+q2 Planes......Page 391
27: Beutelspacher’s Construction of Line Parallelisms......Page 392
1. Extension of Beutelspacher’s Theorem......Page 393
2. Applications of Beutelspacher’s Construction......Page 396
28: Johnson Partial Parallelisms......Page 399
1. Isomorphisms......Page 403
1.2. Infinite Johnson Parallelisms......Page 414
2. The Derived Parallelisms......Page 415
2.1. Characterization......Page 417
Part 7: Parallelism-Inducing Groups......Page 419
29: Parallelism-Inducing Groups for Pappian Spreads......Page 422
30: Linear and Nearfield Parallelism-Inducing Groups......Page 429
1. Parallelism-Inducing Subgroups of GL(2, q2)......Page 433
2. The General Near.eld Parallelism-Inducing Group......Page 435
3. Isomorphisms of Group-Induced Parallelisms......Page 436
4. Nearfield Parallelism-Inducing Groups......Page 438
5. Finite Regular Nearfield Groups......Page 441
6. m-Parallelisms......Page 443
31: General Parallelism-Inducing Groups......Page 450
1. The Isomorphisms of Kantor-Knuth Type Parallelisms......Page 455
2. Finite Regulus-Inducing Elation Groups......Page 458
3. Finite General Parallelism-Inducing Groups......Page 462
4. Minimal Linear Parallelism-Inducing Groups......Page 465
5. Relative Linear Parallelism-Inducing Groups......Page 466
6. Existence of Minimal Groups......Page 473
7. Determination of the Minimal Groups......Page 476
8. General Isomorphisms from Minimal Groups......Page 479
9. Isomorphic Kantor-Knuth Parallelisms......Page 482
10. Even Order......Page 485
11. Derived Parallelisms......Page 487
Part 8: Coset Switching......Page 489
1. E-and Desarguesian Switches......Page 491
2. The Switches of Desarguesian Spreads......Page 493
3. Coset Switching again with Desarguesian Spreads......Page 496
4. An Upper Bound......Page 499
5. Upper Bound, Even Order......Page 502
33: Parallelisms over Ordered Fields......Page 506
34: General Elation Switching......Page 513
0.3. Main Theorem on General Elation Switching......Page 516
1. Deficiency One Transitive Groups......Page 518
2. Switching Theorem over Ordered Fields......Page 522
35: Dual Parallelisms......Page 527
1. The Isomorphisms of the Dual Parallelisms......Page 532
Part 9: Transitivity......Page 540
36: p-Primitive Parallelisms......Page 542
1. Biliotti-Jha-Johnson p-Primitive Theorem......Page 543
4. Johnson\'s Classification of 2-Transitive Parallelisms......Page 549
2. Transitive Parallelisms of PGL(2r–1, 2)......Page 551
3. Isomorphisms of the Transitive Parallelisms......Page 555
1. BJJDJM-Classification Theorem......Page 557
1.1. When G Admits an Elation......Page 559
3. The Full Group......Page 572
39: Doubly Transitive Focal-Spreads......Page 578
1. Johnson-Montinaro; Elementary Abelian Case......Page 580
Part 10: Appendices......Page 585
40: Open Problems......Page 587
1. Non-standard Groups and Non-standard Parallelisms......Page 588
1.1. Non-standard Parallelisms......Page 589
41: Geometry Background......Page 603
42: The Klein Quadric......Page 607
1. The Thas-Walker Construction......Page 608
1. Subgroups of PSL(2, q)......Page 610
2. The Lists of Mitchell and Hartley......Page 611
4. Primitive Subgroups of GammaL(4, q)......Page 613
5. Aschbacher\'s Theorem......Page 614
6. Guralnick-Penttila-Praeger-Saxl Theorem......Page 616
7. O\'Nan-Scott Theorem......Page 617
9. Johnson-Montinaro t-Transitive Theorem......Page 618
44: The Diagram......Page 623
Bibliography......Page 624