Quotient-local powering-invariant subgroup

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Definition

Suppose $G$ is a group and $H$ is a normal subgroup of $G$. Let $\varphi :G\to G/H$ be the corresponding quotient map. We say that $H$ is a quotient-local powering-invariant subgroup of $G$ if the following equivalent conditions hold:

1. For any $g\in G$ and any natural number $n$ such that there exists a unique $x\in G$ satisfying $x^{n}=g$, we also have that $\varphi (x)$ is the unique element of $G/H$ whose $n^{th}$ power is $\varphi (g)$.
2. For any $g\in G$ and any prime number $p$ such that there exists a unique $x\in G$ satisfying $x^{p}=g$, we also have that $\varphi (x)$ is the unique element of $G/H$ whose $p^{th}$ power is $\varphi (g)$.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
local powering-invariant subgroup	any unique root of an element in the subgroup must be in the subgroup			|FULL LIST, MORE INFO
local powering-invariant normal subgroup	local powering-invariant and normal			|FULL LIST, MORE INFO
quotient-powering-invariant subgroup	if the whole group is powered over a prime, so is the quotient group.			|FULL LIST, MORE INFO
powering-invariant subgroup	if the whole group is powered over a prime, so is the subgroup.			|FULL LIST, MORE INFO
powering-invariant normal subgroup	powering-invariant and a normal subgroup.			|FULL LIST, MORE INFO
quotient-torsion-freeness-closed subgroup	if the whole group is $p$-torsion-free for some prime $p$, so is the subgroup				|FULL LIST, MORE INFO