# Derived subgroup is quotient-powering-invariant in nilpotent group

From Groupprops

## Contents

## Statement

Suppose is a nilpotent group. Then, the derived subgroup is a quotient-powering-invariant subgroup of . In other words, if is powered over a prime number , so is the abelianization of (defined as the quotient of 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

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