Self-centralizing subgroup

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:


 * $$C_G(H) \le H$$
 * $$Z(H) = C_G(H)$$

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.

Stronger properties

 * Weaker than::CC-subgroup
 * Weaker than::Centralizer-free 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
For a complete list of examples of self-centralizing subgroups, refer:

Category:Instances of self-centralizing subgroups

Metaproperties
If a subgroup is self-centralizing in the whole group, it is also self-centralizing in every intermediate subgroup.

If $$H$$ is a self-centralizing subgroup of $$G$$, and $$K \le G$$ is a subgroup containing $$H$$, then $$K$$ is also a self-centralizing subgroup of $$G$$.

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