How do the elements of the set $(\mathbb{Q}\backslash\{0\},\cdot)/(\mathbb{Q}^{+},\cdot)$ look like?

Question

I have already asked that Kind of Question previously. I have the Problem that if I have to answer a Question like this than I can Always find a homomorphism for which the desired quotientgroup is the kernel of. And then can apply the Isomorphic Theorem. However I don't know how this helps me to answer the Question how the Elements look like.

For example:

For the above Question my Approach is like that:

I want to find a group homomorphism for which we have

$\phi:(\mathbb{Q}\backslash\{0\},\cdot)\rightarrow(G,\cdot_g)\wedge \ker(\phi)=(\mathbb{Q}^{+},\cdot)$

I can for example let $(G,\cdot_g)$ be $(\mathbb{Z}/2\mathbb{Z},+)$

And then $\phi(x)=\begin{cases}[0],x>0\\ [1],x<0\end{cases}$

Applying the Isomorphtheorem yields $(\mathbb{Q}\backslash\{0\},\cdot)/(\mathbb{Q}^{+},\cdot)\cong (\mathbb{Z}/2\mathbb{Z},+)=\text{Im}(\phi)$

What can I say now if I pick a $x\in (\mathbb{Q}\backslash\{0\},\cdot)$ ?

The Question was literaly to determine the set $(\mathbb{Q}\backslash\{0\},\cdot)/(\mathbb{Q}^{+},\cdot)$.

Answer 1

This quotient group precisely contains two elements (closets), namely, $(1)Q^+$ and $(-1) Q^+$.