Derived subgroup is quotient-powering-invariant in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group. Then, the derived subgroup $$G' = [G,G]$$ is a quotient-powering-invariant subgroup of $$G$$. In other words, if $$G$$ is powered over a prime number $$p$$, so is the abelianization of $$G$$ (defined as the quotient of $$G$$ by its derived subgroup).

Related facts

 * Derived subgroup is quotient-powering-faithful in nilpotent group
 * Nilpotent group is powered over a prime iff its abelianization is

Facts used

 * 1) uses::Derived subgroup is divisibility-closed in nilpotent group
 * 2) uses::Divisibility-closed implies powering-invariant
 * 3) uses::Derived subgroup is normal
 * 4) uses::Normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication

Proof
The proof follows directly by combining Facts (1)-(4). When using Fact (4), note that since $$G$$ is nilpotent, it equals its own hypercenter, so any subgroup is contained in the hypercenter, hence in particular the derived subgroup is contained in the hypercenter.