Tactical dismantling

A tactical decomposition (engl .: tactical decomposition ) is in the finite geometry is a partitioning of the point and the block quantity of a 2- block plan in point and block classes such that each consisting of one of these point classes and one of these block classes pair with the induced incidence forms a tactical configuration . Such a breakdown can be viewed as a generalization of the resolution of a block plan : Unlike in the case of a resolution in which only the block set is partitioned into (generalized "parallels") groups, so that here too the original point set with each of the block classes (groups ) forms a tactical configuration, a tactical decomposition generally also divides the number of points into several point classes .

Definitions

Tactical dismantling

Let be a block plan, be a partition of the point set and a partition of the block set . One calls a tactical decomposition of , if any of the incidence structures ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$ ${\displaystyle 2-(v,k,\lambda )}$ ${\displaystyle \{{\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{d}\}}$ ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle \{{\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{d},{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle {\mathcal {I}}}$

${\displaystyle ({\mathfrak {p}}_{j},{\mathfrak {B}}_{i},I_{\mathrm {Ind} });\;1\leq j\leq d,1\leq i\leq c}$

with the particular induced incidence is a tactical configuration . That means in detail: ${\displaystyle I_{\mathrm {Ind} }=I\cap \left({\mathfrak {p}}_{j}\times {\mathfrak {B}}_{i}\right)}$

There are nonnegative integers with the properties: ${\displaystyle \rho _{ji},\kappa _{ji};\;1\leq j\leq d,1\leq i\leq c}$

1. Exactly blocks go through each point of and ${\displaystyle {\mathfrak {p}}_{j}}$ ${\displaystyle \rho _{ji}}$ ${\displaystyle {\mathfrak {B}}_{i}}$
2. there are exactly points on each block of . ${\displaystyle {\mathfrak {B}}_{i}}$ ${\displaystyle \kappa _{ji}}$ ${\displaystyle {\mathfrak {p}}_{j}}$

Parameters of a tactical decomposition

• The following terms are agreed for tactical dismantling:

${\displaystyle m_{i}=|{\mathfrak {B}}_{i}|,n_{j}=|{\mathfrak {p}}_{j}|;\;1\leq j\leq d,1\leq i\leq c}$,

the amounts hot spot classes , the amounts are called block classes of decomposition. The numbers are called the parameters of the tactical decomposition . ${\displaystyle {\mathfrak {p}}_{j}}$ ${\displaystyle {\mathfrak {B}}_{i}}$ ${\displaystyle \rho _{ji},\kappa _{ji},m_{i},n_{j}}$

Relationships between the parameters of the decomposition

Be a tactical breakdown with the parameters of the block plan . Then: ${\displaystyle \{{\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{d},{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle \rho _{ji},\kappa _{ji},m_{i},n_{j}}$ ${\displaystyle 2-(v,k,\lambda )}$ ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$

1. For each is ${\displaystyle j\in \{1,\ldots ,d\}}$ ${\displaystyle \sum _{i=1}^{c}\rho _{ji}\cdot \kappa _{ji}=\lambda \cdot n_{j}+r-\lambda .}$
2. For everyone with is ${\displaystyle j,h\in \{1,\ldots ,d\}}$ ${\displaystyle j\neq h}$ ${\displaystyle \sum _{i=1}^{c}\rho _{ji}\cdot \kappa _{hi}=\lambda \cdot n_{h}.}$

In addition, the following applies:

${\displaystyle b-v\geq c-d\geq 0.}$

The following sentence by Block and Kantor states that with each tactical breakdown the number of point classes can be at most as large as the number of block classes and that with symmetrical 2-block plans a breakdown is only possible if these class numbers are equal:

Be a tactical breakdown of the block plan . Then: ${\displaystyle \{{\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{d},{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle 2-(v,k,\lambda )}$ ${\displaystyle {\mathcal {I}}}$

1. ${\displaystyle c\geq d}$ and
2. Is symmetrical, so is ${\displaystyle {\mathcal {I}}}$ ${\displaystyle c=d.}$

The proof of the second statement from the first follows simply from the fact that:

If there is a symmetrical 2-block plan and a tactical breakdown, then there is a tactical breakdown of the dual block plan ! ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$ ${\displaystyle \{{\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{d},{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c},{\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{d}\}}$ ${\displaystyle {\mathcal {I}}=({\mathfrak {B}},{\mathfrak {p}},I^{-1})}$

Examples

Trivial decompositions

Each block plan allows the following two trivial tactical decompositions: ${\displaystyle 2-(v,k,\lambda )}$ ${\displaystyle {\mathcal {I}}=({\mathfrak {p}},{\mathfrak {B}},I)}$

1. ${\displaystyle c=1,d=1,{\mathfrak {p}}_{1}={\mathfrak {p}},{\mathfrak {B}}_{1}={\mathfrak {B}},\rho _{11}=r=b_{1},\kappa _{11}=k=v_{1}}$; here both partitions are trivial.
2. ${\displaystyle c=b=b_{0},d=v=v_{0}}$, whereby each point class contains exactly one point and each block class contains exactly one block, the "classes" are each numbered like their single element. With this partitioning and with this numbering applies

${\displaystyle \rho _{ji}=\kappa _{ji}={\begin{cases}1&((p_{j},B_{i})\in I)\\0&((p_{j},B_{i})\not \in I)\end{cases}}.}$

Resolutions as decompositions

Each resolution of a block plan corresponds to the special tactical breakdown with the parameters ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle \{{\mathfrak {p}}_{1}={\mathfrak {p}},{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle \rho _{1i}=\rho _{i},\kappa _{1i}=k;\;1\leq i\leq c.}$

Orbit decompositions

If G is an automorphism group of the block diagram , i.e. a subgroup G of the full automorphism group , then the point trajectories and the block trajectories of the operations of G are on the point or block set, then is a tactical decomposition of . ${\displaystyle {\mathcal {I}}}$ ${\displaystyle G\leq \mathrm {Aut} ({\mathcal {I}})}$ ${\displaystyle \{{\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{d}\}}$ ${\displaystyle \{{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle \{{\mathfrak {p}}_{1},\ldots ,{\mathfrak {p}}_{d},{\mathfrak {B}}_{1},\ldots ,{\mathfrak {B}}_{c}\}}$ ${\displaystyle {\mathcal {I}}}$

Orbital decomposition is probably the most important case of decomposition. It plays an important role both in the construction of new block plans through group expansion (of suitable automorphism groups) and in the classification of block plans and their (full) automorphism groups. Tactical decompositions are therefore also of a certain importance for the classification of finite simple groups : for example, the sporadic math groups are full automorphism groups in Witt's block diagrams .

The two above mentioned trivial decompositions can (usually) be understood as special orbital decompositions:

1. The first trivial decomposition with only one point and block class arises as a special path decomposition through the operation of the full automorphism group , provided that it operates at least simply transitive on the point set and the block set. ${\displaystyle c=1,d=1}$ ${\displaystyle G=\mathrm {Aut} ({\mathcal {I}})}$
2. The second trivial decomposition with all one-element classes arises as a special orbit decomposition through the operation of the one-group . ${\displaystyle c=b,d=v}$ ${\displaystyle 1\leq \mathrm {Aut} ({\mathcal {I}})}$

literature

Articles on individual questions

• Richard E. Block: On the orbits of collineation groups . In: Mathematical Journal . tape 96 , 1967, p. 33–49 ( full text (PDF; 927 kB) [accessed on August 5, 2012]).
• RGR Harris: On automorphisms and resolutions of designs . Dissertation at the University of London. 1975.
• WM Kantor: Automorphism groups of designs . In: Mathematical Journal . tape 109 , 1969, p. 246-252 .
• CW Norman: A characterization of the Mathieu group M ₁₁ . In: Mathematical Journal . tape 106 , 1968, pp. 162-166 .
• H. Beker: On strong tactical decompositions . In: Journal of the London Mathematical Society . tape 16 , 1977, pp. 191-1196 .

Textbooks

• Albrecht Beutelspacher : Introduction to finite geometry I . Block plans. Bibliographical Institute, Mannheim / Vienna / Zurich / New York 1982, ISBN 3-411-01632-9 , 5. Resolutions and decompositions.
• Thomas Beth , Dieter Jungnickel , Hanfried Lenz : Design Theory . BI Wissenschaftsverlag, Mannheim 1986, ISBN 0-521-33334-2 .

Individual evidence

1. ↑ Beutelspacher (1982)
2. ↑ Beth, Jungnickel, Lenz (1986)
3. ↑ Beker (1977)
4. ↑ Beutelspacher (1982), Lemma 5.2.1
5. ↑ Beutelspacher (1982), sentence 5.2.5
6. ↑ Block (1964)
7. ^ Cantor (1969)
8. ↑ Beutelspacher (1982) sentence 5.2.2
9. ↑ Beutelspacher (1982), p. 213