Epicentral series members are completely divisibility-closed in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group. Then, the members of the epicentral series of $$G$$ (with the exception of the zeroth member, the trivial subgroup) are all  proves property satisfaction of::completely divisibility-closed subgroups of $$G$$. Explicitly, this means that if $$p$$ is a prime number such that $$G$$ is $$p$$-divisible, then for any member $$H$$ of the epicentral series of $$G$$ (other than the trivial subgroup), every $$p^{th}$$ root of an element of $$H$$ is in $$H$$.

In particular, the epicenter of a nilpotent group is a completely divisibility-closed subgroup. Note that if the epicenter happens to be the trivial subgroup, then that subgroup is also completely divisibility-closed.

Similar facts

 * Upper central series members are completely divisibility-closed in nilpotent group
 * Annihilator of divisibility-closed subgroup under bihomomorphism is completely divisibility-closed
 * Complete divisibility-closedness is strongly intersection-closed