Centralizer of divisibility-closed subgroup is completely divisibility-closed in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$H$$ is a divisibility-closed subgroup of $$G$$. Then, the centralizer $$C_G(H)$$ of $$H$$ in $$G$$ is a completely divisibility-closed subgroup of $$G$$.

In particular, this shows that the property of being a divisibility-closed subgroup of nilpotent group is a centralizer-closed subgroup property, and also that the property of being a completely divisibility-closed subgroup of nilpotent group is a centralizer-closed subgroup property.

Related facts

 * Upper central series members are completely divisibility-closed in nilpotent group: In fact, the stated fact here can be viewed as a generalization of the fact that upper central series members are completely divisibility-closed.
 * Kernel of a bihomomorphism implies completely divisibility-closed: The ideas used in the proof of the fact on the current page are a generalization and application of the kernel of a bihomomorphism idea.
 * Kernel of a multihomomorphism implies completely divisibility-closed: The ideas used in the proof of the fact on the current page are a generalization and application of the kernel of a multihomomorphism idea.