Dembowski-Wagner theorem

The set of Dembowski-Wagner is one of the classic theorems from the mathematical sub-area of the finite geometry , which in the transition field between combinatorics and finite geometry is located. The sentence goes back to the two mathematicians Peter Dembowski and Ascher Wagner and formulates a number of criteria according to which a symmetrical block plan can be understood as a projective space .

Formulation of the sentence

Given a symmetrical - block diagram with , the incidence relation with the member relation is identical. For the order of be there and . ${\displaystyle 2{\text{-}}(v,k,\lambda )}$ ${\displaystyle {\mathcal {D}}=({\mathfrak {p}},{\mathfrak {B}},\in )}$ ${\displaystyle {\mathfrak {B}}\subsetneqq {\tbinom {\mathfrak {p}}{k}}}$ ${\displaystyle n=k-\lambda }$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle n>1}$ ${\displaystyle \lambda >1}$

Then are equivalent :

(B1) is an incidence structure isomorphic , which is formed by the points and the hyperplanes of a finite projective space together with the element relation as the incidence relation, where the blocks and the hyperplanes correspond to one another. ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$

(B2) Every straight line from intersects every block. ${\displaystyle {\mathcal {D}}}$

(B3) There are exactly points on every straight line from . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle 2+{\frac {n-1}{\lambda }}}$

(B4) Any three non-collinear points of incline always with the same number of blocks. ${\displaystyle {\mathcal {D}}}$

(B5) Each level of is contained in exactly blocks. ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\frac {\lambda (\lambda -1)}{n+\lambda -1}}}$

Explanations and Notes

1. In a projective space, a hyperplane is a maximum real subspace . A hyperplane is thus characterized in that they alone in the projective space itself as a subspace included , but not with this identical is thereby of no third subspace comprises is.
2. A straight line from is a real subset of which arises from two different points from . For this purpose , the intersection is formed over all blocks that contain both and . One calls the straight line through and and writes or similar. One then also says that and lie on the straight line . ${\displaystyle g}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle P,Q}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \lambda }$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle g=PQ}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle g}$
3. It is said that a straight line intersects a block when a with exist incidentally point which is on the straight line is located; ma W. if is. ${\displaystyle g}$ ${\displaystyle B\in {\mathfrak {B}}}$ ${\displaystyle B}$ ${\displaystyle g}$ ${\displaystyle g\cap B\neq \emptyset }$
4. Collinear points are characterized by the fact that they lie on a straight line (then necessarily clearly defined) .
5. Each plane is made up of three different non-collinear points of . Such three points then form a triangle . Just as with the straight lines , the intersection of all those blocks that contain it is formed for this triangle and the plane spanned by is obtained . One writes for it briefly or similar. ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \epsilon }$ ${\displaystyle P,Q,R}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle P,Q,R}$ ${\displaystyle \epsilon }$ ${\displaystyle \epsilon =PQR}$
6. The numbers given above can also be found in the literature in a different but equivalent representation. Because of the parameter conditions and the symmetry property of u. a .: ${\displaystyle {\mathcal {D}}}$
   1. ${\displaystyle v=2n+\lambda +{\frac {n(n-1)}{\lambda }}}$.
   2. ${\displaystyle {\frac {v-\lambda }{k-\lambda }}={\frac {b-\lambda }{r-\lambda }}=1+{\frac {v-1}{k}}=2+{\frac {n-1}{\lambda }}}$.
   3. ${\displaystyle {\frac {\lambda (\lambda -1)}{k-1}}={\frac {\lambda (\lambda -1)}{n+\lambda -1}}={\frac {k(\lambda -1)}{v-1}}={\frac {\lambda k-r}{v-1}}}$.
7. The above illustration shows that is a divisor of and that for reasons of integer numbers there are at least different points on each straight line . ${\displaystyle \lambda }$ ${\displaystyle n-1}$ ${\displaystyle 3}$
8. For some authors, the Dembowski-Wagner sentence is also allowed. In this version of the theorem, the possibility that a projective plane is isomorphic is covered. The conditions mentioned in this version of the sentence are essentially the above without (B5). However, the case in the original work by Dembowski and Wagner is expressly excluded. ${\displaystyle \lambda =1}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \lambda =1}$
9. In the original work by Dembowski and Wagner, other equivalent conditions are mentioned under which a symmetrical block plan can be understood as a projective space. However, these are often not mentioned at all or only in passing in the current literature. These are transitivity requirements for the associated automorphism group , such as its transitive operation on the set of triangles of . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$
10. There are several generalizations of the Dembowski-Wagner theorem. In one of these, a simple block diagram is initially assumed and without requiring symmetry from the outset. The symmetry then results at the same time as the other conditions. Another generalization is made below. ${\displaystyle {\mathcal {D}}}$

Generalization according to Kantor

With regard to the question of the inclusion of finite projective planes in Dembowski-Wagner's theorem, the following result by William Kantor is of interest, which brings this question into the context of matroid theory and thereby generalizes the theorem. The result of Kantor says (in short):

The symmetrical block plans, the blocks of which can be understood as hyperplanes of matroids, coincide with the finite projective geometries. ${\displaystyle 2}$

Here a generalized hyperplane concept comes into play. One understands that is a matroid with associated shell operator under a hyperplane one under real closed subset of which this property with respect to maximum is. ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle {\mathcal {M}}}$

This applies more precisely:

The following conditions are equivalent for the symmetrical block plan with and : ${\displaystyle 2{\text{-}}(v,k,\lambda )}$ ${\displaystyle {\mathcal {D}}=({\mathfrak {p}},{\mathfrak {B}},\in )}$ ${\displaystyle \lambda \in \mathbb {N} }$ ${\displaystyle {\mathfrak {B}}\subsetneqq {{\mathfrak {p}} \choose k}(k\geq 3)}$

(K1) represents the set of hyperplanes of a matroid defined on . ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle {\mathfrak {p}}}$

(K2) Either is

${\displaystyle \lambda =1}$and is regarded as one on projective plane defined order , the set of lines with coinciding ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle n=k-\lambda }$ ${\displaystyle {\mathfrak {B}}}$

      or

it is and is to be understood as a projective space that is defined on and whose set of hyperplanes coincides with . ${\displaystyle \lambda >1}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle {\mathfrak {B}}}$

literature

• Thomas Beth , Dieter Jungnickel , Hanfried Lenz : Design Theory . Bibliographical Institute, Mannheim / Vienna / Zurich 1985, ISBN 3-411-01675-2 .
• Albrecht Beutelspacher : Introduction to Finite Geometry II. Projective Spaces . Bibliographisches Institut, Mannheim / Vienna / Zurich 1983, ISBN 3-411-01648-5 ( MR0670590 ).
• PJ Cameron , JH van Lint : Designs, Graphs, Codes and their Links (=  London Mathematical Society Student Texts . Volume 22 ). Cambridge University Press, Cambridge (et al.) 1991, ISBN 0-521-42385-6 .
• Peter Dembowski : Finite Geometries (=  results of mathematics and their border areas . Volume 44 ). Springer Verlag, Berlin / Heidelberg / New York 1968.
• P. Dembowski, A. Wagner: Some characterizations of finite projective spaces . In: Arch. Math . tape 11 , 1960, pp. 465-469 ( MR0143095 ).
• Daniel R. Hughes, Fred C. Piper: Design Theory . Cambridge University Press, Cambridge (et al.) 1985, ISBN 0-521-25754-9 .
• WM Kantor : 2-Transitive designs. In: Marshall Hall, Jr. , JH van Lint (Ed.): Combinatorics: proceedings of the Advanced Study Institute on Combinatorics held at Nijenrode Castle, Breukelen, the Netherlands, July 8-20, 1974 Part 3. (=  Mathematical Center Tracts ). 2nd, revised edition. tape 57 . Mathematisch Centrum, Amsterdam 1975, ISBN 90-6196-101-7 , p. 44-97 ( MR0376382 ).
• William M. Kantor : Characterizations of finite projective and affine spaces . In: Canad. J. Math . tape 1 , 1969, p. 64-75 ( MR0236040 ).
• ES Lander: Symmetric Designs: An algebraic Approach (=  London Mathematical Society Lecture Note Series . Volume 74 ). Cambridge University Press, Cambridge (et al.) 1983, ISBN 0-521-28693-X .
• DJA Welsh : Matroid Theory (=  LMS Monographs . Volume 8 ). Academic Press, London (et al.) 1976, ISBN 0-12-744050-X .

References and comments

1. ↑ Some characterizations of finite projective spaces by Peter Dembowski and Ascher Wagner
2. ^ Dembowski-Wagner: Arch. Math . tape 11 , p. 465 ff .
3. ↑ Cameron: p. 8 ff.
4. ↑ Welsh: pp. 205 ff.
5. ↑ In the underlying item in the archive of mathematics , Volume 11, 1960, Dembowski and Wagner called a symmetric block plan a -space (Engl. -Space ). ${\displaystyle \lambda }$ ${\displaystyle \lambda }$
6. ^ Hughes-Piper: p. 79 ff.
7. ↑ Lander: pp. 16, 24 ff.
8. ↑ ^{a b} Dembowski-Wagner: Arch. Math . tape 11 , p. 465 .
9. ^ Cantor: Combinatorics. Part 3 . S. 70-71 .
10. ↑ Beth-Jungnickel-Lenz: p. 583.
11. ↑ Beutelspacher: p. 18.
12. ↑ Beth-Jungnickel-Lenz: p. 580.
13. ^ Dembowski: p. 67.
14. ↑ The hyperplanes of the matroid clearly define its structure, since they are reversibly and uniquely linked to the circles of the dual matroid by forming a complement ; see. Welsh: pp. 35-39. ${\displaystyle {\mathcal {M}}^{*}}$
15. ^ Cantor: Can. J. Math . tape 21 , p. 64 ff .
16. ^ Welsh: p. 208.
17. ↑ In the sense specified above!
18. ↑ Please note that for projective planes, straight lines and hyperplanes coincide.