My question has its root from the following one.
Consider the following groups: $$\begin{cases}G_{1}=\{x+y\sqrt3|x,y \in \mathbb Q,x^2-3y^2=1 \}\\ G_{2}=\left\{\begin{pmatrix}x&3y\\ y&x\\\end{pmatrix}|x,y \in \mathbb Q,x^2-3y^2=1 \right\}\end{cases} $$
What are all the group isomorphisms between those two groups? I was trying to use the fact that $\mathbb Q(\sqrt{3})$ is a field extension of $\mathbb Q$... without success.