# Difference between revisions of "Weakly closed conjugacy functor"

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(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 the [[center]] <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. |

## Revision as of 02:10, 15 January 2012

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

## 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 the center , .

- 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.