Here is a theorem From Artin's algebra and then use of it. Here $O$ denotes all orthogonal operators(motions which leave origins fixed). I wonder, why can we change coordinates in $G$? Is it a matter of isomorphism between motions which fix an origin point $O$ and some point $p$?
Yes. If $p\in\mathbb R^3$ and if $G$ is a group such that $p$ is fixed by each element of $G$, let $T_p\colon\mathbb R^3\longrightarrow\mathbb R^3$ be the map defined by $T_p(v)=v+p$. Then, for each $g\in G$, ${T_p}^{-1}\circ g\circ T_p$ leaves the origin $O$ fixed and the groups $G$ and $\{{T_p}^{-1}\circ g\circ T_p\,|\,g\in G\}$ are isomorphic.