Self-centralizing subgroup: Difference between revisions

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]
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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:

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

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

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.
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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.

@@ Line 21: / Line 21: @@
 ===Stronger properties===
-* [[CC-subgroup]]
+* [[Weaker than::CC-subgroup]]
-* [[Centralizer-free subgroup]]
+* [[Weaker than::Centralizer-free subgroup]]
-* [[Self-normalizing subgroup]]
+* [[Weaker than::Self-normalizing subgroup]]
-==Metaproperties==
+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===
-{{trim}}
+* [[Subgroup containing the center]]
-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.