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From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | | | | |
RANDOM SUBGROUP PROPERTY: Fully characteristic subgroup: A subgroup such that any endomorphism of the whole group takes the subgroup to within itself.

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 3 Examples 4 Metaproperties 4.1 Intermediate subgroup condition 4.2 Upward-closedness 4.3 Join-closedness 5 Testing 5.1 GAP code

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:

• 
• Z(H) = C_G(H)

Relation with other properties

Stronger properties

• CC-subgroup
• Centralizer-free subgroup
• 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

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

Category:Instances of self-centralizing subgroups

Metaproperties

Intermediate subgroup condition

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the 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 is a subgroup containing H, then K is also a self-centralizing subgroup of G.

Join-closedness

This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property
View a complete list of join-closed subgroup properties

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

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