# 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.

## 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.