Difference between revisions of "Weakly closed conjugacy functor"

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(Definition)
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# Either of these equivalent:
 
# 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>.
 
#* 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>.
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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 <math>W(P)</math>, <math>W(P) = W(Q)</math>.
 
# Either of these equivalent:
 
# 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>.
 
#* 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 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.
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===Equivalence of definitions===
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{{further|[[equivalence of definitions of weakly closed conjugacy functor]]}}

Revision as of 02:11, 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:
    • There exists a p-Sylow subgroup P such that, for every p-Sylow subgroup Q containing W(P), W(P) = W(Q).
    • For every p-Sylow subgroup P, and for every p-Sylow subgroup Q containing W(P), W(P) = W(Q).
  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.

Equivalence of definitions

Further information: equivalence of definitions of weakly closed conjugacy functor