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Let $G$ be the group of rigid motions of a cube. We aim to prove that the order of $G$, denoted as $|G|$, is 24.

Assume each vertex of the cube is labeled as $v_1, v_2, \ldots, v_8$. Consider sending vertex $v_1$ to position $p_k$. Then, vertex $v_2$ can only be relocated to any of the three positions adjacent to $p_k$. Once the positions for $v_1$ and $v_2$ have been determined, the positions for all other vertices are uniquely determined, as rigid motions preserve distances and angles. This arrangement results in a total of $8 \cdot 3 = 24$ possibilities, as we have $8$ choices of where to send $v_1$ and $3$ choices for $v_2$.

Is this proof sufficiently rigorous? Any tips?

Micah Zarin
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  • This is fine, except for the fact that someone might want more detailed justification for why the positions of the remaining vertices are uniquely determined. If you know the orbit/stabilizer theorem, I would use that directly. – Ted Shifrin Dec 30 '23 at 18:28

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