Algebra Solutions: A cube has 24 orientation-preserving isometries

Let $G$ be the group of rigid motions (i.e., orientation preserving isometries) of a cube in $\mathbb{R}^3$ . Show that $|G| = 24$ .

For reference, consider the following diagram of a cube.

Let $\theta$ be an orientation-preserving isometry of the cube; that is, if the vertices of a face, read clockwise from outside the figure, are $WXYZ$ , then $\theta(W) \theta(X) \theta(Y) \theta(Z)$ are the vertices of the corresponding face, read clockwise from outside the figure, of the isometric copy.
There are 8 possibilities for $\theta(1)$ . Once $\theta(1)$ is chosen, there are 3 possibilities for $\theta(2)$ . Once $\theta(2)$ is chosen, $\theta(3)$ is determined by orientation, and so $\theta(4)$ is determined. The rest of $\theta$ is then uniquely determined by orientation. Thus there are $8 \cdot 3 = 24$ possibilities for $\theta$ , all distinct.

Algebra Solutions

A cube has 24 orientation-preserving isometries

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