## Finite GeometriesReihentext + Finite Geometries From the reviews: "Such a vast amount of information as this book contains can only be accomplished in 375 pages by a very economical style of writing... it enables one to have a good look at the forest without being too detracted by the individual trees... The author deserves unstinting praise for the skill, energy, and perseverance which he devoted to this work. The finished product confirms what his many earlier contributions to the subject of finite geometry have already indicated, namely, that he is an undisputed leader in his field." Mathematical Reviews "Finite Geometries" is a very important area of finite mathematics characterized by an interplay of combinatorial, geometric, and algebraic ideas, in which research has been very active and intensive in recent years... makes it clear how large is the field covered by the author in his book. The material is selected most thoroughly, and the author made an effort to collect all that seems to be relevant in finite geometries for the time being... Dembowski's work will be a basic reference book of this field, and it will be considered as a base of the future research... Altogether this is a very well-produced monograph." Publicationes Mathematicae Debrecen 10, tom 16. |

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### Contents

1 Basic concepts | 1 |

12 Incidence preserving maps | 8 |

13 Incidence matrices | 17 |

14 Geometry of finite vector spaces | 23 |

2 Designs | 56 |

22 Embeddings and extensions | 69 |

23 Automorphisms of designs | 78 |

24 Construction of designs | 92 |

52 Planes of type IV | 228 |

53 Planes of type V | 236 |

54 Planes of types I and II | 246 |

6 Inversive planes | 252 |

62 Combinatorics of finite inversive planes | 262 |

63 Automorphisms | 268 |

64 The known finite models | 273 |

7 Appendices | 281 |

3 Projective and affine planes | 115 |

32 Combinatorics of finite planes | 137 |

33 Correlations and polarities | 151 |

34 Projectivities | 157 |

4 Collineations of finite planes | 169 |

42 Collineation groups | 178 |

43 Central collineations | 187 |

44 Groups with few orbits | 207 |

5 Construction of finite planes | 219 |

72 Hjelmslev planes | 291 |

73 Generalized polygons | 300 |

74 Finite semiplanes | 305 |

Bibliography | 318 |

Dictionary | 367 |

369 | |

371 | |

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### Common terms and phrases

2-group affine geometry affine plane algebra Amer arbitrary automorphism group axis Baer subplane balanced incomplete block Bose Canad central collineations collineation group condition contains coordinatized cosets cyclic defined denote desarguesian desarguesian planes difference sets distinct points doubly transitive dual dually elations elements exists finite inversive planes finite planes finite projective planes fixed point flag transitive following result Hadamard hence Higman Hjelmslev plane implies incidence matrix incidence structure incomplete block designs integer inversive plane involution isomorphic Lemma Luneburg Math matrices nearfield nontrivial normal subgroup number of points orbits orthogonal Ostrom ovoid parallel classes parameters partial design permutation group Pickert planar plane of order point and block point sets point-line pair polarity prime power Proc projective design projective geometry proof proved quadric quasifield satisfying Satz Section semifield sharply transitive Steiner triple systems subspaces tactical configuration tactical decomposition Theorem translation plane unique