Paley's theorem
The Paley theorem , named after the English mathematician Raymond Paley , is a mathematical theorem about the construction of Hadamard block plans using the methods of group theory . As such, it lies in the transition field between combinatorics , geometry and algebra .
Block plans, which can be constructed according to Paley's theorem, are sometimes also referred to as Paley block plans (English Paley designs ) or Paley-Hadamard 2 block plans (English Paley-Hadamard 2 designs ).
Formulation of the sentence
For a prime power of the shape to a natural number, the following always applies:
- (I) There is a - block diagram , that is, a symmetric block diagram with the parameters , , , .
- (II) The associated incidence structure can be constructed in the following way:
- The Galois field to which it belongs is chosen as the point set of ; that is, one chooses the body elements as the points of the incidence structure.
- For the construction of the block system one starts from the multiplicative group of the Galois body and considers here the subgroup of squares , that is . Then you bet .
- The incidence relation is the element relation , so .
Examples of Paley block plans
The two smallest examples of Paley block diagrams are those for the two prime numbers and .
It follows for on the -Blockplan whose geometric structure of the Fano plane corresponds. The subgroup of squares of is described above .
For results on the block plan. The subset of the squares of is here .
Further examples result from other articles in the category: Block plan :
Notes on the proof of the theorem
The proof of Paley's theorem can be carried out with the help of Fisher's inequality and the fact that a special permutation group exists which operates twice homogeneously on .
As it turns out, the block system can also be described in another way, namely as a set of - images of everything , i.e. in form .
The permutation group is obtained from the above subgroup by considering those permutations which have the form where and are fixed elements. All these permutations, provided with the usual concatenation , then form .
It can now be shown that the subgroup has the order , while the order results for the permutation group . So it has odd order and, according to Lagrange's theorem, does not contain an element of order 2 . Hence is , from which then the 2-fold homogeneity of follows.
Related result
Another result about Hadamard block plans goes back to Raymond Paley:
- For each prime power of the figure a Hadamard block plan exists with the parameters , , , , so a symmetrical -Blockplan.
This result shows, for example, the existence of the following Hadamard block plans:
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 I. Block plans . Bibliographical Institute, Mannheim / Vienna / Zurich 1982, ISBN 3-411-01632-9 . MR0670590
- Peter Dembowski : Finite Geometries (= results of mathematics and their border areas . Volume 44 ). Springer Verlag, Berlin / Heidelberg / New York 1968.
- Daniel R. Hughes, Fred C. Piper: Design Theory . Cambridge University Press, Cambridge u. a. 1985, ISBN 0-521-25754-9 .
- Konrad Jacobs , Dieter Jungnickel: Introduction to combinatorics (= de Gruyter textbook ). 2nd, completely revised and expanded edition. de Gruyter, Berlin a. a. 2004, ISBN 3-11-016727-1 .
- Heinz Lüneburg : combinatorics (= elements of mathematics from a higher point of view . Volume 6 ). Birkhäuser Verlag, Basel / Stuttgart 1971, ISBN 3-7643-0548-7 . MR0335267
- REAC Paley : On orthogonal matrices . In: J. Math. Phys. Mass. Inst. Tech. tape 12 , 1933, pp. 311-320 .
References and comments
- ↑ A. Beutelspacher: Introduction to finite geometry . 1982, p. 104-108 .
- ^ H. Lüneburg: combinatorics . 1971, p. 75 ff .
- ↑ T. Beth, D. Jungnickel, H. Lenz: Design Theory . 1985, p. 70 ff., 262, 264 .
- ^ DR Hughes, FC Piper: Design Theory . 1985, p. 107 ff .
- ↑ K. Jacobs, D. Jungnickel: Introduction to combinatorics . 2004, p. 251 ff .
- ^ P. Dembowski: Finite Geometries . 1968, p. 97 .
- ^ DR Hughes, FC Piper: Design Theory . 1985, p. 107, 180 .
- ↑ So .
- ↑ T. Beth, D. Jungnickel, H. Lenz: Design Theory . 1985, p. 262, 264 .
- ↑ Because of the differences in the presentation in the associated main article, note the reference to the Singer cycle .
- ↑ All prime powers of gestalt with a base prime also always provide Paley block plans . For example, for the prime powers, one sees that there is a block plan, a block plan and also a block plan. See also
- ↑ The essential step in the proof is to show that the identical mapping of any two-element subset can be fixed, that is, for and the equation always results; s. A. Beutelspacher: Introduction to finite geometry . 1982, p. 106 . And also H. Lüneburg: combinatorics . 1971, p. 79 .
- ↑ K. Jacobs, D. Jungnickel: Introduction to combinatorics . 2004, p. 252 .
- ↑ T. Beth, D. Jungnickel, H. Lenz: Design Theory . 1985, p. 70-72 .