My question deals with the action of $SO(2)$ on $R^2$. I need to determine whether it is transitive/2-transitive/faithful/free or not. I know it's not transitive, because we can take any two points with different lengths, and there aren't any matrices that will transfer $x_1$ to $x_2$ since orthogonal matrices preserve norm. It is not 2-transitive since it is not transitive. In addition, it is not free since $(0,0)$ is a fixed point for every matrix. However, I couldn't determine what about faithfulness. I would really appreciate any help. Thanks in advance.
2026-03-27 07:50:50.1774597850
Is the action of $SO(2)$ on $R^2$ faithful?
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