Finite geometry

The finite geometry is the part of the geometry that describes explored "classical " finite geometric structures, namely finite affine and projective geometries and their generalizations and finite. The structures themselves, with which this branch of geometry and combinatorics involved are referred to as " finite geometries ".

The properties of finite incidence structures are common today in the field of finite geometry examined, one starts as a rule of such structures, to which a geometric underlying motivation, for example, finite incidence geometry. Typical cases of geometric motivation are the axioms " by two points is exactly one line " or " by three points - on a sphere - is exactly a circle."

Block diagrams are the typical objects of investigation of modern finite geometry, including finite geometries typical. If a classical finite geometry is considered as described below as incidence structure ( rank 2 geometry), any finite, at least two-dimensional affine and projective geometry is a 2- block schedule, so far, the term " block plan" a common generalization of the terms " finite affine geometry "and" finite projective geometry ". The theory of block designs is also known as Design Theory (English: design theory ) refers. This term originates from the statistical design of experiments, which leads to applications of finite geometry in some non- mathematical areas.

An important mathematical application have classical finite geometries and their generalizations in the theory of groups and there especially for the classification of finite simple groups, since it has been shown that many simple groups, for example, all groups of Lie type clearly presented as automorphism groups of finite projective geometries can be. On generalized geometries operate the five sporadic Mathieu groups: they are the full automorphism groups of five specific Wittschen block plans.

Classical finite geometries

With the axiomatization of (real two-and three -dimensional ) geometry around the turn of the 20th century, mainly by Hilbert's system of axioms of Euclidean geometry, the question of finite models for minimal axiom systems of affine and projective geometry which, where already had been previously studied in special cases, for example, by Gino Fano. It had been shown that at least three-dimensional geometries are always desarguessch. As for finite geometries, the set of Pappus and the set of Desargues are equivalent ( expressed algebraically: because by the theorem of Wedderburn every finite skew field a commutative multiplication has ), all finite, at least three-dimensional classical geometries as affine or projective spaces can be over represent a finite field. In contrast, two-dimensional geometries exist nichtdesarguessche, ie affine and projective planes.

Finite levels

Every affine plane comes from a projective plane ( by slitting the projective plane ) from. Therefore, it is sought after mainly projective planes in the question of the existence of finite levels, the theory of which is transparent, as non-isomorphic affine planes can be derived from the same projective plane, while all the accounts of a projective affine plane are isomorphic to each other. The nichtdesarguesschen levels are usually classified by the Lenz - Barlotti classification, which was developed by Hanfried Lenz and Adriano Barlotti in the 1940s and 1950s. In this classification, which is also used for infinite planes include nichtdesarguesschen finite levels of the bilge - Class I ( levels on real Ternärkörpern ), II ( real Cartesian groups), IV (translation levels on real quasi bodies ) or V (translation levels above real half-bodies ) to. For each of these classes, the existence of finite models could be shown, but there are still many open questions of existence. See to existential questions, the open questions and conjectures to the article projective plane, Latin square, difference set and set of Bruck and Ryser.

Finite geometries of classical geometries

In a classic, even infinite geometries can be defined finite induced incidence structures that may be of interest to the global structure of the initial geometry. Classical configurations belonging to closure sets form such finite incidence structures.

• For example, the full Desargues configuration is in a classical geometry a finite incidence structure with 10 points and 10 lines and a symmetric incidence structure in the following sense: The incidence matrix that describes the structure can be chosen as a symmetric matrix.
• Also a complete square in a projective plane may be used as finite incidence structure having the corner points or the corner points including intersections of the opposite sides ( bias points ) and its connection lines are regarded as blocks. Here, if we add the diagonal points, two, not mutually isomorphic incidence structures arise: A Fano - square or an anti -Fano quadrangle.
• In a finite projective space can be defined by a quadratic set an incidence structure, the points, for example ( certain ) the points on the square size and may be blocks ( some ) tangent to the quadratic set. See for example the generalized quadrangle on a hyperboloid.

Finite geometries as geometries or incidence graph structures

A classic finite geometry includes a finite number of types n, for example, they form a three-dimensional geometry of the type set. This classic concept with a finite but arbitrary number of types that build a flag structure of incidence is determined by the finite - Buekenhout Tits geometries (also called diagram geometries ) generalized.

The combinatorial study of finite geometries mostly deals with rank -2- geometries in the sense of diagram geometry, ie, with incidence structures geometries with exactly two different types. In traditional n-dimensional geometries, these will be the conventional points, on the other hand blocks as the subspaces of a certain dimension d with. These are then incidence structures and even 2 - block plans. Most who considered finite geometries are desarguesch, so n-dimensional affine or projective spaces over a finite field with q elements. These block plans are listed as or. For the nichtdesarguesschen levels are separated with the notations used or, where T is the level koordinatisierender Ternärkörper.

Automorphisms

The automorphisms of a finite incidence structure ( ie a finite rank -2 geometry in the sense of Buekenhout and Tits ) are also referred to as a ( generalized) collineations. Each incidence preserving, bijective self-map is an automorphism of the incidence structure. For classical geometries whose block set is exactly the classical set of lines, these automorphisms are precisely the classical collineations.

Also in general the classical case of a finite geometry and whose blocks are d-dimensional sub-spaces, one ( incidence structural) automorphism is usually also an automorphism in the classic sense (that is, which reflects all subspaces to partial areas of the same type). The only exceptions to this rule are the affine anti - Fano spaces over the residue field (see exceptions to this collineation ). In this respect, in the combinatorial restriction goes (except for geometries with exactly 2 points on each line) lost no significant information on two types of a classical finite geometry.