# Difference between revisions of "Weakly closed conjugacy functor"

(New page: {{conjugacy functor property}} ==Definition== Suppose <math>G</math> is a finite group, <math>p</math> a prime number, and <math>W</math> a [[defining ingredient::conjugacy funct...) |
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==Definition== | ==Definition== | ||

− | Suppose <math>G</math> is a [[finite group]], <math>p</math> a [[prime number]], and <math>W</math> a [[defining ingredient::conjugacy functor]] on <math>G</math> with respect to <math>p</math>. We say that <math>W</math> is '''weakly closed''' in <math>G</math> with respect to <math>p</math> if <math>W(P)</math> is a [[defining ingredient::weakly closed subgroup]] of <math>P</math>. | + | Suppose <math>G</math> is a [[finite group]], <math>p</math> a [[prime number]], and <math>W</math> a [[defining ingredient::conjugacy functor]] on <math>G</math> with respect to <math>p</math>. We say that <math>W</math> is '''weakly closed''' in <math>G</math> with respect to <math>p</math> if the following equivalent conditions are satisfied: |

+ | |||

+ | # Either of these equivalent: | ||

+ | #* There exists a <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math> such that <math>W(P)</math> is a [[defining ingredient::weakly closed subgroup]] of <math>P</math> relative to <math>G</math>. | ||

+ | #* For every <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math>, <math>W(P)</math> is a weakly closed subgroup of <math>P</math> relative to <math>G</math>. | ||

+ | # Either of these equivalent: | ||

+ | #* There exists a <math>p</math>-[[Sylow subgroup]] <math>P</math> such that, for every <math>p</math>-Sylow subgroup <math>Q</math> containing <math>W(P)</math>, <math>W(P) = W(Q)</math>. | ||

+ | #* For every <math>p</math>-[[Sylow subgroup]] <math>P</math>, and for every <math>p</math>-Sylow subgroup <math>Q</math> containing <math>W(P)</math>, <math>W(P) = W(Q)</math>. | ||

+ | # Either of these equivalent: | ||

+ | #* There exists a <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math> such that for any <math>p</math>-Sylow subgroup <math>Q</math> of <math>G</math> containing <math>W(P)</math>, <math>W(P)</math> is a [[normal subgroup]] of <math>Q</math>. | ||

+ | #* For every <math>p</math>-[[Sylow subgroup]] <math>P</math> of <math>G</math>, it is true that for any <math>p</math>-Sylow subgroup <math>Q</math> of <math>G</math> containing <math>W(P)</math>, <math>W(P)</math> is a [[normal subgroup]] of <math>Q</math>. | ||

For instance, a [[p-normal group]] is a group in which the conjugacy functor that arises by taking the [[center]] is weakly closed. | For instance, a [[p-normal group]] is a group in which the conjugacy functor that arises by taking the [[center]] is weakly closed. | ||

+ | |||

+ | ===Equivalence of definitions=== | ||

+ | |||

+ | {{further|[[equivalence of definitions of weakly closed conjugacy functor]]}} | ||

+ | |||

+ | ==Equivalence relation induced on the set of Sylow subgroups== | ||

+ | |||

+ | Given a weakly closed conjugacy functor <math>W</math> for a prime <math>p</math>, we obtain an equivalence relation on the set <math>\operatorname{Syl}_p(G)</math> of all <math>p</math>-Sylow subgroups of <math>G</math>. The equivalence relation is as follows: two <math>p</math>-Sylow subgroups <math>P,Q</math> are equivalent if they satisfy the above equivalent conditions, for instance, <math>W(P) = W(Q)</math> (this is the equivalent formulation that makes it easiest to see that the relation is reflexive, symmetric, and transitive). | ||

+ | |||

+ | This equivalence relation partitions the set <math>\operatorname{Syl}_p(G)</math> into equivalence classes. It further turns out that all equivalence classes have the same size, because the conjugation with <math>G</math> permutes them transitively. Moreover: | ||

+ | |||

+ | * The equivalence classes are parametrized by the conjugacy class of <math>W(P)</math>. The number of such equivalence classes is <math>[G:N_G(W(P))]</math> and the size of each equivalence class is <math>[N_G(W(P)):N_G(P)]</math>. | ||

+ | * The equivalence class corresponding to a particular <math>W = W(P)</math> is characterized as precisely those <math>p</math>-Sylow subgroups of <math>G</math> that contain <math>W</math>. | ||

+ | * By the [[congruence condition on index of subgroup containing Sylow-normalizer]], both the number of orbits and the size of each orbit are congruent to 1 modulo <math>p</math>. | ||

+ | |||

+ | Two extreme cases are of interest: | ||

+ | |||

+ | * The case that the equivalence relation has only one equivalence class, which means that <math>W(P) = W(Q)</math> for all <math>P,Q \in \operatorname{Syl}_p(G)</math>, or equivalently, the subgroup <math>W(P)</math> is inside <math>O_p(G)</math>, the [[p-core]]. This is equivalent to <math>W(P)</math> being a [[normal subgroup]] of <math>G</math>. For more, see [[conjugacy functor that gives a normal subgroup]]. | ||

+ | * The case that the equivalence relation has equivalence classes all of size one, i.e., this is the case that <math>N_G(W(P)) = N_G(P)</math>. | ||

+ | |||

+ | ==Relation with other properties== | ||

+ | |||

+ | ===Stronger properties=== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||

+ | |- | ||

+ | | [[Weaker than::conjugacy functor that gives a normal subgroup]] || || || || {{intermediate notions short|weakly closed conjugacy functor|conjugacy functor that gives a normal subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::strongly closed conjugacy functor]] || || || || {{intermediate notions short|weakly closed conjugacy functor|strongly closed conjugacy functor}} | ||

+ | |} |

## Latest revision as of 20:31, 8 July 2013

This article defines a property that can be evaluated for a conjugacy functor on a finite group. |View all such properties

## Contents

## Definition

Suppose is a finite group, a prime number, and a conjugacy functor on with respect to . We say that is **weakly closed** in with respect to if the following equivalent conditions are satisfied:

- Either of these equivalent:
- There exists a -Sylow subgroup of such that is a weakly closed subgroup of relative to .
- For every -Sylow subgroup of , is a weakly closed subgroup of relative to .

- Either of these equivalent:
- There exists a -Sylow subgroup such that, for every -Sylow subgroup containing , .
- For every -Sylow subgroup , and for every -Sylow subgroup containing , .

- Either of these equivalent:
- There exists a -Sylow subgroup of such that for any -Sylow subgroup of containing , is a normal subgroup of .
- For every -Sylow subgroup of , it is true that for any -Sylow subgroup of containing , is a normal subgroup of .

For instance, a p-normal group is a group in which the conjugacy functor that arises by taking the center is weakly closed.

### Equivalence of definitions

`Further information: equivalence of definitions of weakly closed conjugacy functor`

## Equivalence relation induced on the set of Sylow subgroups

Given a weakly closed conjugacy functor for a prime , we obtain an equivalence relation on the set of all -Sylow subgroups of . The equivalence relation is as follows: two -Sylow subgroups are equivalent if they satisfy the above equivalent conditions, for instance, (this is the equivalent formulation that makes it easiest to see that the relation is reflexive, symmetric, and transitive).

This equivalence relation partitions the set into equivalence classes. It further turns out that all equivalence classes have the same size, because the conjugation with permutes them transitively. Moreover:

- The equivalence classes are parametrized by the conjugacy class of . The number of such equivalence classes is and the size of each equivalence class is .
- The equivalence class corresponding to a particular is characterized as precisely those -Sylow subgroups of that contain .
- By the congruence condition on index of subgroup containing Sylow-normalizer, both the number of orbits and the size of each orbit are congruent to 1 modulo .

Two extreme cases are of interest:

- The case that the equivalence relation has only one equivalence class, which means that for all , or equivalently, the subgroup is inside , the p-core. This is equivalent to being a normal subgroup of . For more, see conjugacy functor that gives a normal subgroup.
- The case that the equivalence relation has equivalence classes all of size one, i.e., this is the case that .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

conjugacy functor that gives a normal subgroup | Strongly closed conjugacy functor|FULL LIST, MORE INFO | |||

strongly closed conjugacy functor | |FULL LIST, MORE INFO |