Difference between revisions of "Weakly closed conjugacy functor"

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(Equivalence relation induced on the set of Sylow subgroups)
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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).
 
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 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>.
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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:
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* 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>.
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* 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>.
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* 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:
 
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]]. The equivalent conditions are explored at [[Equivalence of normality and characteristicity conditions for conjugacy functor]].
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* 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.
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* 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>.

Revision as of 20:26, 8 July 2013

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

Equivalence relation induced on the set of Sylow subgroups

Given a weakly closed conjugacy functor W for a prime p, we obtain an equivalence relation on the set \operatorname{Syl}_p(G) of all p-Sylow subgroups of G. The equivalence relation is as follows: two p-Sylow subgroups P,Q are equivalent if they satisfy the above equivalent conditions, for instance, W(P) = W(Q) (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 \operatorname{Syl}_p(G) into equivalence classes. It further turns out that all equivalence classes have the same size, because the conjugation with G permutes them transitively. Moreover:

  • The equivalence classes are parametrized by the conjugacy class of W(P). The number of such equivalence classes is [G:N_G(W(P))] and the size of each equivalence class is [N_G(W(P)):N_G(P)].
  • The equivalence class corresponding to a particular W = W(P) is characterized as precisely those p-Sylow subgroups of G that contain W.
  • 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 p.

Two extreme cases are of interest:

  • The case that the equivalence relation has only one equivalence class, which means that W(P) = W(Q) for all P,Q \in \operatorname{Syl}_p(G), or equivalently, the subgroup W(P) is inside O_p(G), the p-core. This is equivalent to W(P) being a normal subgroup of G. 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 N_G(W(P)) = N_G(P).