Difference between revisions of "Self-centralizing subgroup"

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(Stronger properties)
(Definition)
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===Symbol-free definition===
 
===Symbol-free definition===
  
A [[subgroup]] of a [[group]] is said to be '''self-centralizing''' if it contains its own centralizer in the whole group.
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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
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* Its center equals its centralizer in the whole group
  
 
===Definition with symbols===
 
===Definition with symbols===
  
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''self-centralizing''' if:
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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''self-centralizing''' if it satisfies the following equivalent conditions:
  
<math>C_G(H) \le H</math>
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* <math>C_G(H) \le H</math>
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* <math>Z(H) = C_G(H)</math>
  
 
==Relation with other properties==
 
==Relation with other properties==

Revision as of 08:01, 31 March 2007

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

Metaproperties

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

The trivial subgroup and the whole group are clearly self-centralizing.

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.