conjugacy classes of the special orthogonal group $SO(2)$

Question

I'm doing some research in Group Theory and have come across conjugacy classes. In general I can determine the conjugacy classes for most groups.

However the conjugacy classes of the special orthogonal group $SO(2)$ is causing me difficulties. Can someone describe the structure of the conjugacy classes of this group?

Answer 1

The group $SO(2,\mathbb R)$ is abelian. Therefore, the conjugacy class of each $g\in SO(2,\mathbb R)$ is $\{g\}$.