Derived subgroup is quotient-powering-invariant in nilpotent group

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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

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.