A question on SO(3) equivariance

[wzm2256]

Thanks for the amazing work! I wish to ask two questions regarding the SO(3) equivariance.

If I understand it correctly:

1. The element in $SE(3)_0^N$ is like $((r_1, t_1), ... (r_n, t_n))$ where $\sum t =0$ as defined in the paragraph above prop.3.5. However, $SE(3)_0^N$ is not even a group when $N>1$ (when $N=1$, $SE(3)_0^N$ is just $SO(3)$), because the mutiplication is not even closed. In other words, it is not a Lie group as claimed in the paper, right?
2. A related problem is in Eqn 6, where the diffusion term is PB(t), where P is matrix depending on X, My question is how to derive the reverse process. The paper claims to use Prop 2.1, but it is different from Eqn 6, because there is no matrix P there.

Thanks for your help!