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$SE(3)$ is the representation for both translation and rotation in 3D space, whereas $SO(3)$ is only the representation for rotations. $\mathbb{R}^3$ is for translations in 3D space. If you only consider 2-dimentional space, then you can simply change 3 to 2, i.e., $SE(2)$, $SO(2)$, $\mathbb{R}^2$. I wouldn't worry too much about it. You can just consider ...

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