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]
- It contains its own centralizer in the whole group
- Its center equals its centralizer in the whole group
Definition with symbols
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
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
For a complete list of examples of self-centralizing subgroups, refer:
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.
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 is a self-centralizing subgroup of , and is a subgroup containing , then is also a self-centralizing subgroup of .
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.
One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.GAP-codable subgroup property
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.