$E(n) = \{(A,x):A \in O(n) \text{ and } \mathbf{x} \in \mathbb{R}^n \}$, where $O(n)$ is the group of real orthogonal $n \times n$ matrices
The point group associated to $p \in \mathbb{R}^n$ and a subgroup $G \subset E(n)$ is the stabalizer of $p$ in $G$. The stabilizer of $p$ in $G$ is the group of elements that fix $p$, i.e. $gx=x, \forall g \in G$. So I think the point group of $G$ associated to $p \in \mathbb{R}^n$ is the subgroup of elements $(A,b)$ such that, \begin{equation} Ap + b = p \end{equation} Would this consist of elements of the form $(A,0)$ where $A$ is a rotation matrix by $2 \pi n$?
And then I think the translation subgroup $T$ is the subgroup of $G$ with elements of the form $(I,x)$. Wouldn't this make $G/T$ the set of matrices $(A,0)$ (not necessarily rotation matrices)?
How would you come up with a bijection between these two?