Difference between revisions of "Self-centralizing subgroup"

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===Stronger properties===
 
===Stronger properties===
  
* [[CC-subgroup]]
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* [[Weaker than::CC-subgroup]]
* [[Centralizer-free subgroup]]
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* [[Weaker than::Centralizer-free subgroup]]
* [[Self-normalizing subgroup]]
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* [[Weaker than::Self-normalizing subgroup]]
  
==Metaproperties==
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Under additional conditions:
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* In any group, a [[maximal among Abelian subgroups]]
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* In a [[supersolvable group]] or [[nilpotent group]], [[maximal among Abelian normal subgroups]]
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* In a solvable group, the [[Fitting subgroup]]
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===Weaker properties===
  
{{trim}}
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* [[Subgroup containing the center]]
  
The trivial subgroup and the whole group are clearly self-centralizing.
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==Metaproperties==
  
 
{{intsubcondn}}
 
{{intsubcondn}}
  
 
If a subgroup is self-centralizing in the whole group, it is also self-centralizing in every intermediate subgroup.
 
If a subgroup is self-centralizing in the whole group, it is also self-centralizing in every intermediate subgroup.
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{{upward-closed}}
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If <math>H</math> is a self-centralizing subgroup of <math>G</math>, and <math>K \le G</math> is a subgroup containing <math>H</math>, then <math>K</math> is also a self-centralizing subgroup of <math>G</math>.
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{{join-closed}}
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Since any subgroup containing a self-centralizing subgroup is self-centralizing, a join of any nonempty collection of self-centralizing subgroups is again self-centralizing.

Revision as of 14:48, 14 July 2008

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

Symbol-free definition

A subgroup of a group is said to be self-centralizing if it satisfies the following equivalent conditions:

  • It contains its own centralizer in the whole group
  • Its center equals its centralizer in the whole group

Definition with symbols

A subgroup H of a group G is said to be self-centralizing if it satisfies the following equivalent conditions:

  • C_G(H) \le H
  • Z(H) = C_G(H)

Relation with other properties

Stronger properties

Under additional conditions:

Weaker properties

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If a subgroup is self-centralizing in the whole group, it is also self-centralizing in every intermediate subgroup.

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties

If H is a self-centralizing subgroup of G, and K \le G is a subgroup containing H, then K is also a self-centralizing subgroup of G.

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

Since any subgroup containing a self-centralizing subgroup is self-centralizing, a join of any nonempty collection of self-centralizing subgroups is again self-centralizing.