Steiner system#The Steiner system S.285.2C 8.2C 24.29

As Witt block plans (also Witt designs, Eng. Witt designs) are referred to in the finite geometry block certain plans that were discovered in 1931 by Robert Daniel Carmichael and 1938, described by Ernst Witt, after which they are named again. It consists first of two 5 - block plans that are referred to as small or large Witt cutter plan. Both are up to isomorphism the only simple 5- block designs with point number 12 (small ) or 24 (large Witt Cutter Plan). The small Witt block plan is a block diagram, as Steiner system; the big one is a block diagram, as a Steiner system.

The importance of small and large Wittschen block plan is - for the discrete mathematics - that they were for decades the only known non- trivial 5 - block plans and are therefore examined in great detail. In group theory, more specifically, for the classification of finite simple groups, are the two 5 - block designs and their derivatives, which are often referred to as Witt block plans of great importance, as the Mathieu groups (named after Émile Léonard Mathieu which are 5 of sporadic simple groups ) are their automorphism groups.

• 2.1 Witt block plans
• 2.2 Incidence parameters of Wittschen block plans
• 2.3 Mathieu groups

Construction

Little Witt Cutter plan

The block diagram can be constructed as a three-fold extension of the affine plane of order 3 ( see the right figure). This entails making some special advantage of this level:

• Each quadrilateral V A is a Fano parallelogram, i.e., the four corners of a rectangle, then two pairs of opposite sides of the six sides parallel to each other and the third pair of opposite sides intersect at the thus uniquely determined bias point that no corner point is. ( The n-gon a set of n points of A is referred to, if not the 3 points are collinear. )
• The amount of the 54 squares in A can be decomposed into three classes as per 18 of four corners, each of said equivalence class that has the following characteristics:

Now the set of points are added three extra points and defines the following types of blocks for the new block Quantity:

This provides for a total of 132 blocks each with 6 points: 12 for the extended straight line ( first type), 12 for the complements of the line, these are the parallel pairs of A ( second type) and the 54 for the extended rectangles (3rd type) and the extended pairs of intersecting straight lines ( 4 type).

The marked incidence structure is a block diagram.

Big Witt Cutter plan

The big Witt block diagram can be constructed as a three-fold expansion of the projective plane of order 4.

Properties

Witt block plans

• Each block diagram is isomorphic to the above-constructed block plan and each has a unique automorphism of sequel to a automorphism of. This sequel is determined that operates as a permutation on the set of square classes described above, and is then continued by. In addition, each block diagram is isomorphic to the derivation of the small Witt block plan at any point x.
• The small Witt - block plan contains exactly 12 Hadamard - sub-block plans.
• Each block diagram is isomorphic to the constructed above block diagram.
• Every block diagram is used to derive, the derivation of the large Witt block plan at any point x isomorphic.
• Each block diagram is used to derive, twice the derivation of the large Witt block plan at any two different points x, y isomorphic.

Incidence parameters of Wittschen block plans

The parameters of a finite incidence structure satisfying a regularity condition, are those of the incidence parameter ( average number of blocks by i arbitrary points ) or ( average score balanced by j any blocks ), the matching of all i -element point sets and j -element block amounts of positive numbers same. When small and large Wittschen 5 - block plan, both of which have as incidence structures of type (5.1 ), these are the parameters and. After each discharge a block parameter satisfies its less regularity condition:

In addition, a dependent only on the score section number can be specified for subsets of a block B if is. In other words, the B and U independent number of blocks that have in common with B exactly all points of U. The following table shows these average figures:

Using these average figures can prove the uniqueness of the Wittschen block plans ( up to isomorphism, as block plans with their respective parameters).

Mathieu groups

The five sporadic Mathieu groups are the full automorphism groups of Witt block plans, where the subscript on the short name corresponds respectively to the subscript of the associated Witt - block schedule, v whose score. All five are simple groups, ie they have no non-trivial normal subgroups. Pure group theory allows the subscript v of Matthieugruppen also describe as the minimum integer v such that operates as a permutation group on, in other words, is the smallest symmetric group, so that there is a Gruppenmonomorphismus. The parameter t of the block plan indicating how many arbitrary points exists in each case a common block, are group- theoretically the maximum Transitivitätsgrad the associated Matthieugruppe to, that is, the group operates as a t -fold, but not fold transitive permutation group on the points the corresponding block plan and can operate more than t- fold transitive and faithful on any lot.