Self-centralizing subgroup: Difference between revisions

Latest revision as of 15:39, 4 April 2010

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]
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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:

$C_{G}(H)\leq H$
$Z(H)=C_{G}(H)$

Note that the term self-centralizing subgroup is often used for an abelian self-centralizing subgroup, i.e., a subgroup that equals, rather than merely contains, its centralizer. This is equivalent to being maximal among abelian subgroups.

Relation with other properties

Stronger properties

Under additional conditions:

In any group, a maximal among Abelian subgroups
In a supersolvable group or nilpotent group, maximal among Abelian normal subgroups
In a solvable group, the Fitting subgroup

Weaker properties

Subgroup containing the center

Examples

For a complete list of examples of self-centralizing subgroups, refer:

Category:Instances of self-centralizing subgroups

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 $K\leq G$ 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.

Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsSelfCentralizing
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

GAP-codable subgroup property

A short piece of GAP code can test whether a subgroup of a group is self-centralizing: the code is available at GAP:IsSelfCentralizing.

@@ Line 5: / Line 5: @@
 ===Symbol-free definition===
-A [[subgroup]] of a [[group]] is said to be '''self-centralizing''' if it contains its own centralizer in the whole group.
+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]] <math>H</math> of a [[group]] <math>G</math> is said to be '''self-centralizing''' if:
+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>
+* <math>Z(H) = C_G(H)</math>
-<math>C_G(H) \le H</math>
+Note that the term '''self-centralizing subgroup''' is often used for an ''abelian'' self-centralizing subgroup, i.e., a subgroup that ''equals'', rather than merely contains, its centralizer. This is equivalent to being [[maximal among abelian subgroups]].
 ==Relation with other properties==
@@ Line 17: / Line 23: @@
 ===Stronger properties===
-* [[CC-subgroup]]
+* [[Weaker than::CC-subgroup]]
-* [[Centralizer-free subgroup]]
+* [[Weaker than::Centralizer-free subgroup]]
-* [[Self-normalizing subgroup]]
+* [[Weaker than::Self-normalizing subgroup]]
+Under additional conditions:
+* In any group, a [[maximal among Abelian subgroups]]
+* In a [[supersolvable group]] or [[nilpotent group]], [[maximal among Abelian normal subgroups]]
+* In a solvable group, the [[Fitting subgroup]]
+===Weaker properties===
+* [[Subgroup containing the center]]
+==Examples==
-==Metaproperties==
+For a complete list of examples of self-centralizing subgroups, refer:
-{{trim}}
+[[:Category:Instances of self-centralizing subgroups]]
-The trivial subgroup and the whole group are clearly self-centralizing.
+==Metaproperties==
 {{intsubcondn}}
 If a subgroup is self-centralizing in the whole group, it is also self-centralizing in every intermediate subgroup.
+{{upward-closed}}
+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>.
+{{join-closed}}
+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.
+==Testing==
+{{GAP code for subgroup property|test = IsSelfCentralizing}}
+A short piece of GAP code can test whether a subgroup of a group is self-centralizing: the code is available at [[GAP:IsSelfCentralizing]].