A question about equivariance to 3D transformations using semi-direct and direct products. in the paper: https://arxiv.org/pdf/2010.02449.pdf the author consider some classes of functions that are invariant to the action of the group $G= \mathbb{R}^3 \rtimes SO(3) \times S_n$ on $\mathbb{R}^{3 \times n}$.
In simple terms, I understand that they're asking some function $f$, taking a point cloud $X \in \mathbb{R}^{3 \times n}$ as input to be equivariant to translations, rotations and permutations.
In general, I know that a semi-direct product is established when we have $H,K \le G$, ($H$ normal), $HK=G$ and $H \cap K = \{e\}$. Then we can define the following map
\begin{align}\phi: H\times K &\rightarrow G
\\  (h,k)&\mapsto hk
\end{align}
to be a group isomorphism by introducing the group operation $(h_1,k_1)\cdot(h_2,k_2):= (h_1k_1h_2k_1^{-1},k_1k_2)$, then we write $G = H \rtimes K$.
So what I don't understand here is, why we need semi-direct product of $\mathbb{R}^3$ and $SO(3)$ and then direct product with $S_n$ to establish equivariance to the aforementioned transformations? Which is the point?
 A: You should read this small section: https://en.wikipedia.org/wiki/Direct_product_of_groups#Semidirect_products. It explains when a group decomposes as a direct product or as a semidirect product. It basically comes down to whether $hk=kh$ for all $h\in H$, $k\in K$ or not. If it helps you, I think of the $\rtimes$ as an inner (or internal) semidirect prouct, and of the $\times$ as an outer (or external) direct product in $\left(\Bbb R^3 \rtimes SO(3)\right) \times S_n$.
$\Bbb R^3 \rtimes SO(3) \cong E^+(3)$ is the group of rigid motions (i.e. orientation-preserving isometries) of $\Bbb R^3$. The semidirect product decomposition expresses that any an isometry can be expressed in a unique way as a rotation followed by a translation. The set of translations is a normal subgroup of $E^+(3)$, but $SO(3)$ isn't, which means we get a semidirect product but not a direct product. In particular, translations don't commute with rotations.
Then $\left(\Bbb R^3 \rtimes SO(3)\right) \times S_n = E^+(3)\times S_n$ means that we consider transformations given by permuting the points followed by moving the whole point cloud with a rigid motion. We use the direct product because:

*

*Our transformations are uniquely given by a point permutation and a rigid motion.

*Point permutations and rigid motions commute with each other.


Another way to think of the difference between direct and semidirect products might be to ask whether the elements in $H$ and $K$ "interact" with each other. Say we looked at $\Bbb R^3 \times SO(3)$ instead of $\Bbb R^3 \rtimes SO(3)$. This would be like having two separate Euclidean spaces and then translating one of them and rotating the other -> no interaction.
