Exercise 1.7.21 (The group of rigid motions of a cube is isomorphic to $S_4$)

Proof. Consider the cube $C=\{A_1,A_2,\dots ,A_8\}$ with vertices

Let $G$ be the group of rigid motions of a cube. Every rigid motion $g\in G$ gives a permutation of the 8 vertices of the cube, thus fixes their gravity center $O=(0,0,0)$, so $g$ is an axial rotation, or $g=e$.

We give an arbitrary numbering of the four long diagonals (pairs of opposite vertices):

Since $g$ is an isometry, $||\stackrel{\to }{g(A_2)g(A_7)}||=||\stackrel{\to }{A_2{A}_7}||=2\sqrt{3}$, hence the pair $\{g(A_2),g(A_7)\}$ is a long diagonal, and it is the same for the other long diagonals. Therefore $G$ acts on the set

of the four long diagonals by

The associate permutation representation is

We search the kernel of $\phi$. $g\in \ker <!--\; FUNCTION\; APPLICATION\; -->(\phi )$ if and only if

so $g$ maps the four lines containing the $4$ long diagonals on themselves.

Assume, for the sake of contradiction, that $g\ne e$. Then $g$ is an axial rotation. If every vertex $A_i$ maps to the opposite vertice, then $g=s_O$ is the symmetry through $O$ ( $g$ and $s_O$ are affine transformations which send at least $4$ non coplanar points on the same images). But $s_0$ is not a rigid motion. Therefore there is some long diagonal $\{A_i,A_j\}$ such that $g(A_i)=A_i,g(A_j)=A_j$. Then the axis of the rotation is $A_i{A}_j$, a long diagonal. The only lines globally invariant by such a rotation are the axis, and perhaps the lines in the orthogonal plane (if $g$ is a rotation of $\pi$ radians). Therefore the 3 others long diagonals, which are fixed by $g$, would be orthogonal to the axis, but the four diagonals are not mutually orthogonal. This contradiction shows that $g=e$, so

Therefore $\phi$ is injective (the action is faithful) and $G\simeq \phi (G)\subset S_G$.

Since $|A|=4$, $S_A\simeq S_4$. We prove in Exercise 1.2.10 that $|G|=24=|S_4|=|S_G|$. Therefore $\phi :G\to S_A$ is bijective: it is an isomorphism, so

(another proof is given in Exercise 7.5.10 of “Cox, Galois Theory”) □