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>.
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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:
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# Either of these equivalent:
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#* 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>.
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#* 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>.
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# Either of these equivalent:
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#* 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>.
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#* 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>.
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# Either of these equivalent:
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#* 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>.
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#* 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 G is a finite group, p a prime number, and W a conjugacy functor on G with respect to p. We say that W is weakly closed in G with respect to p if the following equivalent conditions are satisfied:

  1. Either of these equivalent:
  2. Either of these equivalent:
  3. Either of these equivalent:

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