Derived subgroup is quotient-divisibility-faithful in nilpotent group

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., derived subgroup) always satisfies a particular subgroup property (i.e., quotient-divisibility-faithful subgroup)}
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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.

Related facts

Dual fact

For more on the background, see subgroup-quotient duality for groups.

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

The duality is as follows:

Notion for derived subgroup side	Notion for center side
abelianization	center
derived subgroup	inner automorphism group
nilpotent group	nilpotent group
divisible group for a set of primes	torsion-free group for a set of primes
quotient group for a quotient-divisibility-faithful subgroup	torsion-faithful subgroup

Opposite facts

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

Facts used

1. 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.