Difference set

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A difference set of order n (English: perfect difference set ) is a set of natural numbers in finite geometry , from which a unique projective plane can be generated. James Singer was able to prove in the 1930s that every finite Desarguean plane descends from a difference set. This fact is one of the statements of Singer's theorem , which also says that every finite Desargue's projective geometry has a Singer cycle . It is assumed, but has not yet been proven (2012), that precisely the Desarguean finite levels derive from a set of differences.

Definitions

Let n be a natural number. A set of natural numbers is called a difference set of order n , if holds

  1. contains exactly elements,
  2. every natural number can be written in exactly one way than with

The second condition can be formally weakened. Be the diagonal in . Then the 2nd condition is initially equivalent to the more abstractly formulated condition

(2a) The image is bijective .

Since for a set that contains elements according to the 1st condition , the set of pairs of different numbers always contains elements, the definition set of is always equal to the target set, therefore, for this mapping, surjectivity , injectivity and bijectivity are equivalent requirements and the 2nd condition can through

(2b) "For the differences are pairwise different numbers (in other words: is injective)." Or by
(2c) "Every natural number appears modulo as a difference (in other words: is surjective)."

be replaced.

Reduced amount of differences

  • If a difference set is of order , then the different sets for any such difference set are also.
  • Every set of differences of the order contains exactly two different elements with Then is also such a set of differences.

Singer uses difference sets, which contain 0 and 1 and whose elements are all in , as normal forms for difference sets and describes such a difference set as a reduced difference set (English: reduced perfect difference set ). Beutelspacher and Rosenbaum use quantities that contain 1 and 2 as the normal form and whose elements are all in , without introducing a separate designation. The following applies:

If there is a set of differences of order , then there is also one that contains 0 and 1 (i.e. a reduced set of differences) of order .

Properties and meaning

Projective plane

If a difference set is of order , then the geometry defined as follows is a projective plane of order :

  1. The point set is the set of natural numbers,
  2. the set of straight lines consists of the subsets ,
  3. the incidence relation of is the set theoretic containment relation together with its inverse:

One then says: The projective level defined in this way “comes from the set of differences ”.

Singer cycle, Singer theorem

Let be a collineation on a finite projective geometry. If the points and hyperplanes geometry permuted cyclically, ie in the case of a finite level of order : if for any valid

then the collineation group generated by is called a Singer cycle of geometry, especially of the plane.

The set of Dembowski Hughes Parker states that a group of collineations a projective geometry exactly then to the point amount transitive operates when on the set of hyperplanes acts transitively. It follows that the required properties (1) and (2) for cyclic collineation groups are equivalent on one level .

The following statements are called Singer's Theorem :

  1. Every finite, Desarguean, projective geometry has a Singer cycle. This can be chosen so that it even only consists of projectivities .
  2. A finite projective plane has a Singer cycle if and only if it is isomorphic to a plane derived from a set of differences.

If such a plane is represented by the difference set as described above , then is

a collineation of the order which thus creates a Singer cycle.

Construction of Singer cycles on a Desarguean geometry

Every Desargue's projective geometry of finite order is isomorphic to a - dimensional projective space over a finite body . The coordinate vector space of is a vector space isomorphic to the finite field . The multiplicative group is cyclic , so there is a generating (“primitive”) element of this group with which applies. The image

is a vector space automorphism. After choosing a point base in , this automorphism can be viewed as a coordinate representation of a projectivity . Since it operates transitively on , the projectivity represented thereby also operates transitively on the point set of and therefore generates a Singer cycle of this projective geometry.

Examples

The figure shows the Fano plane and a projectivity c of order 7 (red), which creates a Singer cycle. The points (black) are numbered in such a way that this model of the Fano plane derives from the set of differences , the numbers of the straight lines (blue) are
i from the straight line representation
  • The set is a difference set of order 2, because all the differences of different elements read (modulo 7):
The 7 straight lines of the projective plane for this set of differences are, see also the figure on the right:
The plane is isomorphic to the Fano plane .
  • The sets or are difference sets of order 3 and 4, respectively.
  • The set is a reduced difference set of order 5.
  • Since there are no projective planes for orders 6, 10, 12 and 14, there are also no sets of differences between these orders.
  • The Bruck-Ryser-Chowla theorem provides necessary conditions for the orders of projective planes. Natural numbers that are excluded according to this theorem (sequence A046712 in OEIS ) cannot be orders of a difference set either.

literature

  • Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry . From the basics to the applications (=  Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X ( table of contents [accessed April 1, 2012]).
  • Daniel Hughes, Fred Piper: Projective planes (=  Graduate texts in mathematics . Volume 6 ). Springer, Berlin / Heidelberg / New York 1973, ISBN 3-540-90044-6 .
  • James Singer: A theorem in projective geometry and some applications to number theory . In: Transactions of the American Mathematical Society . tape 43 , no. 3 , 1938, pp. 377–385 ( full text, PDF [accessed April 1, 2012]).

References and comments

  1. a b c d e Beutelspacher & Rosenbaum (2004)
  2. a b c Singer (1938)
  3. In this article, 0 is always counted among the natural numbers .
  4. ↑ Please note that due to the properties of the modulo function mod is always a mapping.
  5. In honor of James Singer see literature, Beutelspacher & Rosenbaum (2004), 2.8
  6. ^ Hughes & Piper (1973)
  7. Beutelspacher & Rosenbaum (2004), Chapter 6
  8. Beutelspacher & Rosenbaum (2004), sentences 2.8.4, 2.8.5