C-closed implies local powering-invariant

Statement
Suppose $$G$$ is a group and $$H$$ is a c-closed subgroup of $$G$$, i.e., $$H$$ occurs as the centralizer of some subset (and hence also of some subgroup) of $$G$$. Then, $$H$$ is a local powering-invariant subgroup of $$G$$: for any natural number $$n$$ and any element $$h \in H$$ such that the equation $$x^n = h$$ has a unique solution $$x \in G$$, we must have $$x \in H$$.

Applications

 * C-closed implies powering-invariant
 * Powering-invariance is centralizer-closed
 * Center is local powering-invariant

Facts used

 * 1) uses::c-closed implies fixed-point subgroup of a subgroup of the automorphism group
 * 2) uses::Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant

Proof
The proof follows directly from Facts (1) and (2).

Journal references

 * , Theorem 13.3, Page 229 (13th page in the paper)