Derived subgroup is quotient-divisibility-faithful in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$G'$$ is its derived subgroup. Then $$G'$$ is a quotient-divisibility-faithful subgroup of $$G$$. More explicitly, if $$p$$ is a prime number such that the quotient group $$G/G'$$, which is also the abelianization of $$G$$, is $$p$$-divisible, then $$G$$ is also $$p$$-divisible.

Dual fact
The dual fact to this is dual::center is torsion-faithful in nilpotent group.

The duality is as follows:

Opposite facts

 * Derived subgroup not is quotient-powering-faithful in nilpotent group

Facts used

 * 1) uses::Equivalence of definitions of nilpotent group that is divisible for a set of primes

Proof
The result follows from Fact (1), specifically the (1) iff (2) equivalence within that.