# Thread: iso to the circle group

1. ## iso to the circle group

I need to prove that the special orthogonal group $\displaystyle SO(2,\mathbb{R}) is$ isomorphic to the circle group $\displaystyle S^1$.

I was considering using the case in which

$\displaystyle \varphi :A= \begin{bmatrix} cos \alpha & -sin \alpha \\ sin \alpha & cos \alpha \end{bmatrix} \mapsto (cos \alpha, sin \alpha)$

2. Originally Posted by ux0
I need to prove that the special orthogonal group $\displaystyle SO(2,\mathbb{R}) is$ isomorphic to the circle group $\displaystyle S^1$.

I was considering using the case in which

$\displaystyle \varphi :A= \begin{bmatrix} cos \alpha & -sin \alpha \\ sin \alpha & cos \alpha \end{bmatrix} \mapsto (cos \alpha, sin \alpha)$

As simple as that.

Tonio