Finite geometry

The finite geometry is the part of the geometry , the "classical" finite geometric structures, namely finite affine and projective geometries explored and their finite generalizations and describes. The structures themselves, with which this sub-area of geometry and combinatorics is concerned, are referred to as “finite geometries”.

In general, the properties of finite incidence structures are now being investigated in the field of finite geometry , with structures that are based on a geometric motivation, for example finite incidence geometries, being the starting point . Typical cases of a geometric motivation are the axioms “exactly a straight line goes through two points” or “exactly a circle goes through three points - on a sphere”.

Block plans are the typical objects of investigation in modern finite geometry, including typical finite geometries. If a classical finite geometry is considered as an incidence structure (rank-2 geometry) as described below, every finite, at least two-dimensional affine and projective geometry is a 2-block plan, so the term “block plan” is 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 originally comes from the statistical design of experiments , which leads to applications of finite geometry in some non-mathematical areas.

Classical finite geometries and their generalizations in group theory have an important mathematical application, especially for the classification of finite simple groups , since it has been shown that many simple groups, for example all groups of the Lie type, are clearly represented as automorphism groups of finite projective geometries can be. The five sporadic Mathieu groups operate on generalized geometries : They are the full automorphism groups of five specific Witt block plans .

Classical finite geometries

With the axiomatization of ( real two- and three- dimensional ) geometry at the turn of the 20th century, largely through Hilbert's system of axioms of Euclidean geometry , the question of finite models for the minimal axiom systems of affine and projective geometry was raised had previously been examined in special cases, for example by Gino Fano . It has been shown that at least three-dimensional geometries are always desarguean . As for finite geometries of the set of Pappos and of Desargues set equivalent (expressed algebraically: because according to the set of Wedderburn each finite skew a commutative multiplication has), all finite, at least three-dimensional classical geometries as can affine or projective space via represent a finite body . In contrast, there are non-Desargue's two-dimensional geometries, i.e. affine and projective planes .

Finite levels

Each affine plane descends from a projective plane (by slitting this projective plane). Therefore, the question of the existence of finite planes is predominantly looking for projective planes, the theory of which is clearer, since non-isomorphic affine planes can derive from the same projective plane, while all projective terminations of an affine plane are isomorphic to one another. The non-Desargue plains are typically classified by the Lenz-Barlotti classification developed by Hanfried Lenz and Adriano Barlotti in the 1940s and 1950s. In this classification, which for un used finite levels that nichtdesarguesschen finite levels are one of the Lenz classes I (levels above real Ternärkörpern ), II (via real Cartesian groups ), IV ( translation planes over real quasi bodies ) or V (translation planes over real half-bodies ). The existence of finite models has been shown for each of these classes, but many existential questions are still open. For existential questions, the open questions and the assumptions about them, see the articles Projective Plane , Latin Square , Set of Differences and Theorem by Bruck and Ryser .

Finite geometries from classic geometries

In classic, even un finite geometries, finite be induced incidence structures define that may be of interest to the global structure of the starting geometry. The classical configurations that belong to closure clauses form such finite incidence structures.

For example, the complete Desargues configuration in a classical geometry is a finite incidence structure with 10 points and 10 lines and a symmetrical incidence structure in the following sense: The incidence matrix that describes the structure can be chosen as a symmetrical matrix .
A complete rectangle in a projective plane can also be understood as a finite incidence structure, with the corner points or the corner points including the intersection points of the opposite sides (diagonal points) and their connecting lines as blocks. If you add the diagonal points, two types of incidence structures that are not isomorphic to one another can arise: a Fano quadrangle or an anti-Fano quadrangle .
In a finite projective space an incidence structure can be defined by a quadratic set , whereby the points can be, for example, (certain) points on the quadratic set and the blocks can be (certain) tangent spaces to the quadratic set. See as an example the generalized square on a hyperboloid.

Finite geometries as diagram geometries or incidence structures

A finite number n of types belongs to a classical finite geometry ; these form the type set for a three-dimensional geometry, for example . This classical concept with a finite but arbitrary number of types that build up a flag structure of incidence is generalized by the finite Buekenhout-Tits geometries (also called diagram geometries ). ${\ displaystyle \ Delta = \ {\ mathrm {Point, straight line, plane} \}}$ ${\ displaystyle n \ geq 2}$

The combinatorial investigation of finite geometries mostly deals with rank-2 geometries in the sense of diagram geometry, i.e. with incidence structures , geometries with exactly two different types . In classic n -dimensional geometries this is the part spaces are on one hand the conventional points, on the other hand, as blocks of a predetermined dimension d with . These are then incidence structures and even 2-block plans. Most of the finite geometries considered are desargue, i.e. n -dimensional affine or projective spaces over a finite body with q elements . These block plans are then noted as or . The notations or are occasionally used for the non-Desarguessian levels , where T is a ternary body that coordinates the level. ${\ displaystyle \ Delta = \ {\ mathrm {point, block} \}}$ ${\ displaystyle 1 \ leq d <n}$ ${\ displaystyle \ mathbb {F} _ {q}}$ ${\ displaystyle AG_ {d} (n, q)}$ ${\ displaystyle PG_ {d} (n, q)}$ ${\ displaystyle AG_ {1} (2, T) = AG (2, T)}$ ${\ displaystyle PG_ {1} (2, T) = PG (2, T)}$

Automorphisms

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

Even in the more general classical case of a finite geometry or whose blocks are d- dimensional subspaces, an (incidence structure) automorphism is usually also an automorphism in the classical sense (which therefore maps all subspaces to subspaces of the same type ). The only exceptions to this rule are the affine anti-Fano spaces above the remainder class field (see collineation for these exceptions ). In this respect, no essential information is lost with the combinatorial restriction to two types in a classical finite geometry (except for geometries with exactly 2 points on each straight line). ${\ displaystyle AG_ {d} (n, q)}$ ${\ displaystyle PG_ {d} (n, q)}$ ${\ displaystyle \ mathbb {Z} / 2 \ mathbb {Z}}$

literature

Albrecht Beutelspacher : Introduction to Finite Geometry . I. Block plans. BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1982, ISBN 3-411-01632-9 .
Albrecht Beutelspacher : Introduction to Finite Geometry . II. Projective spaces. BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1983, ISBN 3-411-01648-5 .
Thomas Beth , Dieter Jungnickel , Hanfried Lenz : Design Theory . BI Wissenschaftsverlag, Mannheim 1986, ISBN 0-521-33334-2 .
Lynn Margaret Batten: Combinatorics of Finite Geometries . 2nd Edition. Cambridge University Press, 1997, ISBN 978-0-521-59993-1 .
Peter Dembowski : Finite Geometries. Springer, Berlin / Heidelberg / New York 1968, ISBN 978-3-540-61786-0 .
Peter Dembowski: Combinatorial Properties of Finite Incidence Structures . In: Mathematical Journal . tape 75 , no. 1 , 1961, pp. 256-270 , doi : 10.1007 / BF01211024 .

Web links

Elements of Finite Geometry (website)
Essay on Finite Geometry by Michael Greenberg (PDF file; 138 kB)
Finite geometry (script)

References and comments

↑ Bethe, Jung, Lenz (1986)

↑ Beutelspacher (1982) p. 40: “These designations (the designators for the parameters of a block plan) come from the theory of experimental design, which is one of the sources of finite geometry: is the number of varieties that the blocks and gives the number of replications . " ${\ displaystyle t- (v, k, \ lambda)}$ ${\ displaystyle v}$ ${\ displaystyle b}$ ${\ displaystyle r}$

↑ These minimal axiom systems are described in the articles Affine Geometry and Projective Geometry .

^ Charles Weibel: Survey of Non-Desarguesian Planes . In: Notices of the American Mathematical Society . tape 54 . American Mathematical Society, November 2007, pp. 1294–1303 ( full text (PDF, 702 kB) [accessed December 25, 2011]).

↑ A Desargues configuration is complete if, apart from the collinearities of points assumed or asserted in Desargues' theorem, no further collinearities apply.

[BJP-1] Bethe, Jung, Lenz (1986)