Ambivalence is quotient-closed

This article gives the statement, and possibly proof, of a group property (i.e., ambivalent group) satisfying a group metaproperty (i.e., quotient-closed group property)
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Statement

If $G$ is an ambivalent group, and $H$ is a normal subgroup of $G$, the quotient group $G/H$ is also an ambivalent group.

Related facts

• Rationality is quotient-closed

Proof

Given: An ambivalent group $G$, a normal subgroup $H$ of $G$ with quotient map $\varphi :G\to G/H$, and an element $u\in G/H$.

To prove: There exists an element $v\in G/H$ such that $vuv^{-1}=u^{-1}$.

Proof:

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	Let $g\in G$ be such that $\varphi (g)=u$.		The quotient map $\varphi :G\to G/H$ is, by definition, surjective.		direct.
2	There exists $h\in G$ such that $hgh^{-1}=g^{-1}$.		$G$ is ambivalent.		direct from definition.
3	The element $v=\varphi (h)$ works, i.e., $vuv^{-1}=u^{-1}$.				We start with $hgh^{-1}=g^{-1}$, apply $\varphi$ to both sides, and use the properties of homomorphisms.