Wittscher block plan

As Wittsche block plans (also Witt designs , engl. Witt designs ) certain block plans are called in the finite geometry , which were discovered by Robert Daniel Carmichael in 1931 and described again in 1938 by Ernst Witt , after whom they are also named. There are initially two 5-block plans, which are referred to as the small and  large Wittscher block plan . Except for isomorphism, both are the only simple 5-block plans with the number of points 12 (smaller) or 24 (large Wittscher block plan). Witt's little block plan is a block plan, as a Steiner system ; the large one is a block plan, as a Steiner system . ${\displaystyle \mathrm {W} _{12}}$ ${\displaystyle 5-(12,6,1)}$ ${\displaystyle S(5,6;12)}$ ${\displaystyle \mathrm {W} _{24}}$ ${\displaystyle 5-(24,8,1)}$ ${\displaystyle S(5,8;24)}$

The importance of Witt's small and large block diagram lies - for discrete mathematics - in the fact that for decades they were the only known, non-trivial 5-block diagrams and have therefore been examined in great detail. In group theory , more precisely for the classification of finite simple groups , the two 5-block plans and their derivatives , which are often also referred to as Witt's block plans, are of great importance because the Mathieu groups (named after Émile Léonard Mathieu , are 5 of the sporadic simple groups , ) are their automorphism groups. ${\displaystyle \mathrm {W} _{11},\mathrm {W} _{23},\mathrm {W} _{22}}$ ${\displaystyle \mathbb {M} _{12},\mathbb {M} _{11},\mathbb {M} _{24},\mathbb {M} _{23},\mathbb {M} _{22}}$

construction

Small Wittscher block plan

Geometric construction

The affine plane . ${\displaystyle A=AG_{1}(2,3)}$

The block plan can be constructed as a threefold extension of the affine plane of order 3 (see the figure on the right). One makes use of some special features of this level: ${\displaystyle 5-(12,6,1)}$ ${\displaystyle \mathrm {W} _{12}=({\mathfrak {q}},{\mathfrak {B}},\in )}$ ${\displaystyle A=AG_{1}(2,3)=({\mathfrak {p}},{\mathfrak {G}},\in )}$

• Each square in is a Fano parallelogram , that is, if the four corners of a square are, then two pairs of opposing sides under the six sides are parallel to each other and the third pair of opposing sides intersects at the diagonal point , which is clearly determined by this , which is not a corner point. (A corner is a set of points if none of the 3 points are collinear .) ${\displaystyle v}$ ${\displaystyle A}$ ${\displaystyle p_{1},p_{2},p_{3},p_{4}}$ ${\displaystyle G_{ij}=p_{i}p_{j};\;(1\leq i<j\leq 4)}$ ${\displaystyle d(v)}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle A}$
• The set of 54 squares in can be broken down into three classes of 18 squares each so that each of these equivalence classes has the following properties: ${\displaystyle A}$ ${\displaystyle V_{1},V_{2},V_{3}}$ ${\displaystyle V_{j}}$

1. Each point of is contained in exactly 8 squares , ${\displaystyle A}$ ${\displaystyle V_{j}}$
2. two different points of each lie in exactly 3 squares , ${\displaystyle A}$ ${\displaystyle V_{j}}$
3. every triangle of is contained in exactly one square from . ${\displaystyle A}$ ${\displaystyle V_{j}}$

Now three additional points are added to the point set and the following types of blocks are defined for the new block set : ${\displaystyle q_{1},q_{2},q_{3}\not \in {\mathfrak {p}}}$ ${\displaystyle \left({\mathfrak {q}}={\mathfrak {p}}\cup \{q_{1},q_{2},q_{3}\}\right)}$ ${\displaystyle {\mathfrak {B}}}$

1. For every line G of A let ${\displaystyle G^{*}=G\cup \{q_{1},q_{2},q_{3}\}}$
2. and (these are the points of a pair of parallels of A ) blocks of . ${\displaystyle G^{c}={\mathfrak {p}}\setminus G}$ ${\displaystyle \mathrm {W} _{12}}$
3. For every square v of A with let ${\displaystyle v\in V_{j}}$ ${\displaystyle v^{*}=(v\cup \{q_{1},q_{2},q_{3}\})\setminus \{q_{j}\}}$
4. and blocks of . ${\displaystyle v^{+}=v\cup \{d(v),q_{j}\}}$ ${\displaystyle \mathrm {W} _{12}}$

This results in a total of 132 blocks with 6 points each: 12 for the extended straight lines (1st type), 12 for the complements of the straight lines, i.e. the pairs of parallels of A (2nd type) and 54 each for the extended quadrilaterals (3rd type). Type) and the extended pairs of intersecting lines (4th type). ${\displaystyle \mathrm {W} _{12}}$

The incidence structure defined in this way is a block diagram. ${\displaystyle \mathrm {W} _{12}=({\mathfrak {q}},{\mathfrak {B}},\in )}$ ${\displaystyle 5-(12,6,1)}$

Large Wittscher block plan

Witt's large block plan can be constructed as a threefold extension of the projective level of order 4. ${\displaystyle \mathrm {W} _{24}}$ ${\displaystyle PG_{1}(2,4)}$

properties

Witt block plans

• Each block diagram is isomorphic to the block diagram constructed above and each automorphism of has a unique continuation to an automorphism of . This continuation is determined by operating as a permutation on the set of the quadrilateral classes described above , and then continuing through. In addition, every block plan is isomorphic to the derivation of Witt's little block plan at any point x . ${\displaystyle 5-(12,6,1)}$ ${\displaystyle \mathrm {W} _{12}}$ ${\displaystyle \alpha }$ ${\displaystyle \mathrm {W} _{9}=AG_{1}(2,3)}$ ${\displaystyle {\overline {\alpha }}}$ ${\displaystyle \mathrm {W} _{12}}$ ${\displaystyle \alpha }$ ${\displaystyle \pi \in S_{3}}$ ${\displaystyle V_{1},V_{2},V_{3}}$ ${\displaystyle \alpha (V_{i})=V_{\pi (i)},i\in \{1,2,3\}}$ ${\displaystyle {\overline {\alpha }}(q_{i})=q_{\pi (i)}}$ ${\displaystyle 4-(11,5,1)}$ ${\displaystyle \mathrm {W} _{11}=\mathrm {W} _{12,x}}$
• The small Witt block diagram contains exactly 12 Hadamard - -Unterblockpläne. ${\displaystyle \mathrm {W} _{12}}$ ${\displaystyle 3-(12,3,2)}$
• Each block diagram is isomorphic to the block diagram constructed above . ${\displaystyle 5-(24,8,1)}$ ${\displaystyle \mathrm {W} _{24}}$
• Each block plan is isomorphic to the derivation , the derivation of Witt's large block plan at any point x . ${\displaystyle 4-(23,7,1)}$ ${\displaystyle \mathrm {W} _{23}=\mathrm {W} _{24,x}}$
• Every block plan is isomorphic to the derivation , the double derivative of Witt's large block plan at any two different points x, y . ${\displaystyle 3-(22,6,1)}$ ${\displaystyle \mathrm {W} _{22}=\mathrm {W} _{24,x,y}}$

Incidence parameters of Witt's block plans

The parameters of a finite incidence structure that satisfy a regularity condition are those of the incidence parameters (average number of blocks by any i points) or  (average number of points on any j blocks), which correspond to positive numbers for all i- element sets of points or j- element sets of  blocks same. In the small and large Wittschen 5-block diagram, both of which have the type (5,1) as incidence structures, these are the parameters and . After each derivation, one less block parameter satisfies its regularity condition: ${\displaystyle b_{i}}$ ${\displaystyle v_{j}}$ ${\displaystyle b_{0},b_{1},\ldots b_{5}=1}$ ${\displaystyle v_{0},v_{1}}$

Regular incidence parameters

Block plan	Type as an incidence structure	b 5	b 4	b 3	b 2	b 1 ( r )	b 0 (total number of blocks)	v 2	v 1 ( k )	v 0 (total points)
${\displaystyle \mathrm {W} _{9}\cong AG_{1}(2,3)}$	(2.1)	-	-	-	1	4th	12	-	3	9
${\displaystyle \mathrm {W} _{10}}$	(3.1)	-	-	1	4th	12	30th	-	4th	10
${\displaystyle \mathrm {W} _{11}}$	(4.1)	-	1	4th	12	30th	66	-	5	11
${\displaystyle \mathrm {W} _{12}}$	(5.1)	1	4th	12	30th	66	132	-	6th	12
${\displaystyle \mathrm {W} _{21}\cong PG_{1}(2,4)}$	(2.2)	-	-	-	1	5	21st	1	5	21st
${\displaystyle \mathrm {W} _{22}}$	(3.1)	-	-	1	5	21st	77	-	6th	22nd
${\displaystyle \mathrm {W} _{23}}$	(4.1)	-	1	5	21st	77	253	-	7th	23
${\displaystyle \mathrm {W} _{24}}$	(5.1)	1	5	21st	77	253	759	-	8th	24

In addition, for subsets of a block B, a number of cuts that only depends on the number of points can be specified, if is. In other words, is the number of blocks that are independent of B and U and that have exactly all points of U in common with B. The following table shows these numbers of cuts: ${\displaystyle U\subseteq B\in {\mathfrak {B}}}$ ${\displaystyle u=|U|}$ ${\displaystyle n_{u}=n(B,U)=\left|\{Y\in {\mathfrak {B}}|B\cap Y=U\}\right|}$ ${\displaystyle u\leq k}$ ${\displaystyle n_{u}}$

Cutting numbers

t	k	v 0	n 8	n 7	n 6	n 5	n 4	n 3	n 2	n 1	n 0
2	3	9	-	-	-	-	-	1	0	3	2
3	4th	10	-	-	-	-	1	0	3	2	3
4th	5	11	-	-	-	1	0	3	2	3	0
5	6th	12	-	-	1	0	3	2	3	0	1
2	5	21st	-	-	-	1	0	0	0	4th	0
3	6th	22nd	-	-	1	0	0	0	4th	0	16
4th	7th	23	-	1	0	0	0	4th	0	16	0
5	8th	24	1	0	0	0	4th	0	16	0	30th

With the help of these cutting numbers, one can prove the uniqueness of Witt's block plans (except for isomorphism, as block plans with their respective parameters).

Mathieu groups

The 5 sporadic Mathieu groups are the full automorphism groups of Witt's block plans, whereby the subscript on the short name corresponds to the subscript of the associated Witt block plan, i.e. its number of points v . All five are simple groups, that is, they have none other than the trivial normal divisors . In purely group-theoretical terms, the subscript v of the math groups can also be described as a minimal integer , so that operates on as a permutation group , in other words, is the smallest symmetrical group , so that a group monomorphism exists. The parameter of the block diagram, which specifies for how many arbitrary points a common block exists, specifies the maximum degree of transitivity of the associated math group in terms of group theory, i.e. the group operates as a -fold, but not -fold transitive permutation group on the points of the corresponding math group Block plan and cannot operate transitive and faithful on any set more than -fold. ${\displaystyle \mathbb {M} _{11},\mathbb {M} _{12},\mathbb {M} _{22},\mathbb {M} _{23},\mathbb {M} _{24}}$ ${\displaystyle v}$ ${\displaystyle \mathbb {M} _{v}}$ ${\displaystyle \{1,2,\ldots ,v\}}$ ${\displaystyle S_{v}}$ ${\displaystyle \mathbb {M} _{v}\rightarrow S_{v}}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle t+1}$ ${\displaystyle t}$

Mathieu group	Group order	Block plan	parameter ${\displaystyle t-(v,k,\lambda )}$	Steiner notation
${\displaystyle \mathbb {M} _{11}}$	7920 ${\displaystyle =2^{4}\cdot 3^{2}\cdot 5\cdot 11}$	${\displaystyle \mathrm {W} _{11}}$	${\displaystyle 4-(11,5,1)}$	${\displaystyle S(4,5;11)}$
${\displaystyle \mathbb {M} _{12}}$	95040 ${\displaystyle =2^{6}\cdot 3^{3}\cdot 5\cdot 11}$	${\displaystyle \mathrm {W} _{12}}$	${\displaystyle 5-(12,6,1)}$	${\displaystyle S(5,6;12)}$
${\displaystyle \mathbb {M} _{22}}$	443520 ${\displaystyle =2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11}$	${\displaystyle \mathrm {W} _{22}}$	${\displaystyle 3-(22,6,1)}$	${\displaystyle S(3,6;22)}$
${\displaystyle \mathbb {M} _{23}}$	10200960 ${\displaystyle =2^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 23}$	${\displaystyle \mathrm {W} _{23}}$	${\displaystyle 4-(23,7,1)}$	${\displaystyle S(4,7;23)}$
${\displaystyle \mathbb {M} _{24}}$	244823040 ${\displaystyle =2^{10}\cdot 3^{3}\cdot 5\cdot 7\cdot 11\cdot 23}$	${\displaystyle \mathrm {W} _{24}}$	${\displaystyle 5-(24,8,1)}$	${\displaystyle S(5,8;24)}$

literature

Original article

• Thomas Beth , Dieter Jungnickel : Mathieu Groups, Witt Designs and Golay Codes . In: Geometries and Groups (=  Lecture Notes in Mathematics ). tape 893 . Springer, Berlin / Heidelberg / New York 1981, ISBN 3-540-11166-2 , pp. 157-179 .
• Robert Daniel Carmichael: Tactical Configurations of Rank Two . In: American Journal of Mathematics . tape 53 , 1931, pp. 217-240 , JSTOR : 2370885 .
• Ernst Witt: Mathieu's 5-fold transitive groups . In: Abh. Math. Sem. Univ. Hamburg . tape 12 , 1938, pp. 256-264 , doi : 10.1007 / BF02948947 .

Textbooks

• Thomas Beth, Dieter Jungnickel, Hanfried Lenz : Design Theory . 2nd Edition. BI Wissenschaftsverlag, London / New York / New Rochelle / Melbourne / Sidney 1999, ISBN 0-521-33334-2 , IV: Witt designs and Mathieu groups.
• Albrecht Beutelspacher : Introduction to Finite Geometry . I. Block plans. BI Wissenschaftsverlag, Mannheim / Vienna / Zurich 1982, ISBN 3-411-01632-9 , 2.4: A 5-block plan .

Web links

• Pegg, Ed. Jr., Eric W. Weisstein : Witt design . In: MathWorld (English).
• Eric W. Weisstein : Mathieu Groups . In: MathWorld (English).
• The sporadic groups (producers, subgroups, conjugate classes ...) in the Atlas of Finite Group Representations (English)

Individual evidence

1. ↑ ^{a b} Beutelspacher (1982)
2. ↑ ^{a b c d e} Beth, Jungnickel, Lenz (1999)
3. ↑ Carmichael (1931)
4. ↑ Witt (1938)
5. ↑ Beutelspacher (1982), main clause 2.4.6
6. ↑ A sketch of this construction, which goes back to Witt (1938), can be found in Beth, Jungnickel, Lenz (1999), IV.6.4: Construction
7. ↑ Beth, Jungnickel, Lenz (1999), Corollary IV.2.6
8. ↑ Beth, Jungnickel, Lenz (1999), Lemma IV.4.11
9. ↑ Beth, Jungnickel, Lenz (1999), Theorem IV.5.12