Powering-invariance is centralizer-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., powering-invariant subgroup) satisfying a subgroup metaproperty (i.e., centralizer-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about powering-invariant subgroup |Get facts that use property satisfaction of powering-invariant subgroup | Get facts that use property satisfaction of powering-invariant subgroup|Get more facts about centralizer-closed subgroup property

Statement

Suppose $G$ is a group and $H$ is a powering-invariant subgroup of $G$. Then, the centralizer of $H$ in $G$, i.e., the group $C_{G}(H)$, is also a powering-invariant subgroup of $G$.

Facts used

1. c-closed implies powering-invariant

Proof

By Fact (1), the centralizer of any subgroup in a group is powering-invariant. In particular, the centralizer of a powering-invariant subgroup is powering-invariant.