Centrally closed subgroup

Symbol-free definition
A subgroup of a group is termed centrally closed or a CC-subgroup if the centralizer of any non-identity element of the subgroup lies inside the subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed centrally closed or a CC-subgroup if for any $$h$$ in $$H$$, the group $$C_G(h)$$ lies inside $$H$$.

Stronger properties

 * Malnormal subgroup
 * Frobenius kernel

Weaker properties

 * Self-centralizing subgroup

Metaproperties
If $$H$$ is a CC-subgroup of $$K$$ and $$K$$ is a CC-subgroup of $$G$$, then $$H$$ is a CC-subgroup of $$G$$. The proof of this follows directly from the definitions.

The trivial subgroup is clearly CC, and so is the whole group. Thus, the property of being a CC-subgroup is also a

Note that proper nontrivial CC-subgroups can occur only in a centerless group.

Any subgroup that is CC in the whole group, is also CC in any intermediate subgroup. This is because the centralizer in any intermediate subgroup is contained inside the centralizer in the whole group.