Exercise 1.2.10 (Group of rigid motions of a cube)

Proof. Let $𝒞=\{A,B,C,D,E,F,G,H\}$ a cube (for instance the points of coordinates $(\pm 1,\pm 1,\pm 1)$), and $G$ be the group of rigid motions of $𝒞$.

$G$ acts on the set $𝒞$ by $g\cdot M=g(M)$ $(g\in G,M\in 𝒞)$. Using rotations of angles $\pm \pi ∕2$ and axe $0I$, where $I$ is a centre of a face, we can send $A$ on any vertex. So the action is transitive, there is a unique orbit ${\mathcal{}O}_A$, and $|{\mathcal{}O}_A|=|𝒞|=8$.

The stabilizer $G_A$ of $A$ has three elements: the identity $e$ and the two rotations of angle $\pm 2\pi ∕3$ and axe $\mathit{OA}$. By the fundamental theorem of group actions, $(G:G_A)=|{\mathcal{}O}_A|$, so

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