## Foundations of Translation PlanesAn 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. |

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### Contents

An Overview | 1 |

Andres Theory of Spreads | 9 |

Spreads in PG3 K | 25 |

Partial Spreads and Translation Nets | 39 |

Spreadsets and Partial Spreadsets | 59 |

Vr | 69 |

General Cases | 95 |

Partial Quasifields | 119 |

Structure of Baer Groups | 283 |

Frobenius Complements pPrimitive Collineations and Klein | 297 |

Large Planar Groups | 305 |

Finite Generalized Andre Systems and Nearfields | 317 |

Elation Net Theory | 339 |

BaerElation Theory | 361 |

Semifields | 383 |

Simple TExtensions of Derivable Nets | 401 |

Coordinatization by Partial Quasifields | 135 |

Rational Desarguesian Nets | 155 |

Quasigroups Loops and Nuclei | 163 |

Algebraic Axioms and Autotopisms | 175 |

The Kernel of Spreadsets and Quasifields | 215 |

Hall Systems | 229 |

Spreads in Projective Spaces | 243 |

Kernel Subplanes across Desarguesian Nets | 255 |

Derivation of Finite Spreads | 263 |

Foulsers Covering Theorem | 275 |

Cyclic Semifields | 415 |

Baer Groups on Parabolic Spreads | 421 |

Lifting and Quasifibrations | 437 |

Mixed Tangentially Transitive Planes | 467 |

Maximal Partial Spreads | 481 |

FoulserJohnson SL2 gTheorem | 489 |

APPENDICES | 499 |

527 | |

535 | |

### Common terms and phrases

Abelian Abelian group additive group admits affine plane affine space algebraic automorphism group axes axis Baer group Baer subplane bijection central collineations coaxis collineation group consider coordinatization corollary cosets defined definition denote derivable Desarguesian partial spread Desarguesian plane Desarguesian spread dimension dimension-one elation group elements equivalent Exercise F-space Fix(a Fix(r fixes Foulser-cover GF(q GL(V group G group isomorphic Hence Hom(V homology group incidence structure induces ir(Q K-subspace kernel kernel homologies kernel subplanes leaves invariant lemma Let Q Let TT linear linespreads mapping matrices middle nucleus non-trivial non-zero notation order q2 p-group partial quasifield partial spreadset plane of order prequasifield projective plane projective space Proof proposition pseudoquasifield quadratic quasi-spreadset quasifibration quasigroup rational partial spread regulus Remark right vector space semifield plane skewfield slopeset spread isomorphism spread TT subfield subgroup subpartial subquasifield subspace subspread Suppose Sylow p-subgroup tangentially transitive theorem translation plane unique unit point vector space y-axis