Local powering-invariant and normal iff quotient-local powering-invariant in nilpotent group

Statement
Suppose $$G$$ is a nilpotent group and $$H$$ is a subgroup of $$G$$. The following are equivalent:


 * 1) $$H$$ is a local powering-invariant normal subgroup of $$G$$, i.e., $$H$$ is both a  local powering-invariant subgroup of $$G$$ and a normal subgroup of $$G$$.
 * 2) $$H$$ is a  quotient-local powering-invariant subgroup of $$G$$.

Opposite facts

 * Local powering-invariant and normal not implies quotient-local powering-invariant
 * Center not is quotient-local powering-invariant

Applications

 * Local powering-invariance is quotient-transitive in nilpotent group
 * Upper central series members are local powering-invariant in nilpotent group

Facts used

 * 1) uses::Quotient-local powering-invariant implies local powering-invariant
 * 2) uses::Equivalence of definitions of nilpotent group that is torsion-free for a set of primes
 * 3) uses::Nilpotency is quotient-closed

(2) implies (1)
This follows from Fact (1) (note that normality is definitional).

(1) implies (2)
Given: A nilpotent group $$G$$, a subgroup $$H$$ of $$G$$. $$H$$ is normal and local powering-invariant in $$G$$. An element $$g \in G$$ and a prime number $$p$$ such that the equation $$u^p = g$$ has a unique solution $$u \in G$$. $$\varphi:G \to G/H$$ is the quotient map.

To prove: $$\varphi(u)$$ is the unique $$p^{th}$$ root of $$\varphi(g)$$ in the quotient group $$G/H$$.

Proof: