How to show that these groups are isomorphic? Show that group of all real matrices of form 
$$
\begin{bmatrix}
x & y\\
-y & x
\end{bmatrix} , \qquad (x,y) \ne (0,0)
$$
is isomorphic with/to $\mathbb C \setminus \left\{{0}\right\}$ under complex multiplication?
I know two ways to show isomorphism: 1) finding a homomorphic function 2) writing the multiplication table and comparing.
I think the latter is possible for finite groups.
 A: $\newcommand{\C}{\mathbb{C}}\newcommand{\R}{\mathbb{R}}$What's behind this exercise is the following. 
Consider $\C$ as a vector space over $\R$, with basis $1, i$.
For each $x + i y \in \C$, with $x, y \in \R$, consider the map 
$$
\C \to \C \qquad z \mapsto z \cdot (x + i y).
$$
This map is $\R$-linear, and its matrix with respect to the basis $1, i$ is precisely
$$
\begin{bmatrix}
x & y\\
-y & x
\end{bmatrix}
$$
See if you can get on from here.
Note The matrix is the one above if you consider row vectors. If you consider column vectors, then take the map $z \mapsto z \cdot (x - i y)$, which is still $\R$-linear.
A: You should never need to write out a multiplication table in order to prove that two groups are isomorphic, unless the groups given are actually defined by their multiplication table.  
In this case, the groups are infinite anyway, so you need to find a homomorphic function and show that it's homomorphic and a bijection.  
Some hints to help you find such a function: 


*

*Consider the case $y=0$.  How do the matrices $\begin{pmatrix}x&0\\0&x\end{pmatrix}$ behave under multiplication?  What does that remind you of in the complex numbers?

*What is $\begin{pmatrix}0&-1\\1&0\end{pmatrix}^2$?  What does that remind you of?  

*Using these, what seems like the most natural map from matrices of the form $\begin{pmatrix}x&-y\\y&x\end{pmatrix}$ ($x,y\ne0$) to complex numbers of the form $a+ib$ ($a,b\ne0$)?  Can you prove that that map is a homomorphism and a bijection?  
A: Let $G$ be the group of those matrices with usual matrix multiplication. Define
$$f:G\to \Bbb C\setminus \{0\}$$
$$f\big(\begin{bmatrix}
x & y\\
-y & x
\end{bmatrix}\big)=x+iy$$
Show that it is an isomorphism.
