Question. Let $M = \left\{ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \,\middle\vert\, a, b \in \mathbb R \right\}$.
Show that $(M,+)$ and $(\mathbb C,+)$ are isomorphic.
Show that $(M^{*},*)$ and $(\mathbb C^{*},*)$ are isomorphic.
(Here, $M^{*}$ is defined as excluding the zero-element)
I understand that there exists an isomorphism $\phi:M \rightarrow\mathbb C$ defined by $\begin{bmatrix} \alpha & \beta \\ -\beta & \alpha \end{bmatrix}\mapsto \alpha+i\beta$,
ie. I can show that $\phi(M_{1}+M_{2}) = \phi(M_{1})+\phi(M_{2})$ and $\phi(M_{1}*M_{2}) = \phi(M_{1})*\phi(M_{2})$.
However, I'm unsure how to go about the two subquestions seperately.
Best Answer
Well the map $\phi $ you described has an inverse that maps $a+b i$ to the matrix you have given, this map is also a homomorphism, so the two groups are isomorphic (with respect to addition).
Notice that with the same two maps, the multiplication structure holds as well so they are isomorphic with respect to multiplication.
You've basically done everything but you just need the map
$ \psi(a+b i)=\left( \begin{array}{ccc} a & b \\ -b & a \end{array} \right)$