Let $$G = \left\{\begin{pmatrix} q&0\\a+bi&q\end{pmatrix} \mid q \in \mathbb{Q}^\ast, a,b\in\mathbb{R}\right\}$$ and $$H = \left\{\begin{pmatrix} q&0\\a+ai&q\end{pmatrix} \mid q \in \mathbb{Q},q>0, a\in\mathbb{R}\right\}$$ Which known group is the quotient group $(G, ·)/H$ isomorphic to?
I don't know how to solve this. What I've tried:
$A,B \in G$ belong to the same coset iff $A^{-1}B \in H$. Let $A = \begin{pmatrix} q&0\\a+bi&q\end{pmatrix}$ and $B = \begin{pmatrix} r&0\\c+di&r\end{pmatrix}$. Then $A^{-1} = \begin{pmatrix} 1/q&0\\(-a-bi)/q^2&1/q\end{pmatrix}$ and $A^{-1}B = \begin{pmatrix} r/q&0\\\frac{qc-ra+i(qd-rb)}{q^2}&r/q\end{pmatrix}$
That means that $qc-ra$ should be equal to $qd-rb$. But I don't know how to move from here. Can anyone point me in the right direction or tell me what I'm doing wrong?
Clearly we can identify $(G,\cdot)$ with $(\mathbb{Q}^* \times \mathbb{C},*)$ by
$$ \begin{pmatrix} q & 0 \\ z & q \end{pmatrix} \mapsto (q,z) $$ where $(q,z)*(r,w) = (qr, qw+rz)$. Note that $(\mathbb{Q}^* \times \mathbb{C},*) \simeq (\mathbb{Q}^*,\cdot) \oplus (\mathbb{C},+)$, in fact $(q,z) = (q,0)*(1,\frac{z}{q})$ uniquely and
$$ (\{ (q,0) \in G: q \in \mathbb{Q}^* \},*) \simeq (\mathbb{Q}^*,\cdot) $$ $$ (\{ (1,z) \in G: z \in \mathbb{C} \},*) \simeq (\mathbb{C},+) $$ since $(q,0)*(r,0) = (qr,0)$ and $(1,z)*(1,w) = (1,z+w)$. With this identification
$$ H \simeq (\mathbb{Q}^+,\cdot) \oplus (\mathbb{R}(1+i),+) \simeq (\mathbb{Q}^+,\cdot) \oplus(\mathbb{R},+). $$ Now since $$ a+ib = \frac{1}{2}\Big( (a+b)(1+i)+(a-b)(1-i) \Big), $$ we have $\mathbb{C} \simeq \mathbb{R}(1+i) \oplus \mathbb{R}(1-i) \simeq \mathbb{R}\oplus \mathbb{R}$. Finally
$$ \frac{G}{H} \simeq \frac{(\mathbb{Q}^*,\cdot) \oplus (\mathbb{C},+)}{(\mathbb{Q}^+,\cdot) \oplus (\mathbb{R},+)} \simeq (\{\pm1\} \times \mathbb{R},*). $$ Now $(\{\pm1\} \times \mathbb{R},*) \simeq (\mathbb{R},+)$ by $\varphi$ defined $(a,r) \mapsto ar$, in fact $$ \varphi \Big( (a,r)*(b,s) \Big) = \varphi(ab,as+br) = ab(as+br) = bs + ar = \varphi(a,r)+\varphi(b,s). $$
Therefore $(G/H,\cdot) \simeq (\mathbb{R},+)$.