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. Derived subgroup is divisibility-closed in nilpotent group
2. Divisibility-closed implies powering-invariant
3. Derived subgroup is normal
4. 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.