Algebra Solutions: Find the number of orientation-preserving isometries of an octahedron

Let $G$ be the group of rigid motions (i.e., orientation preserving isometries) of an octahedron in $\mathbb{R}^3$ . Show that $|G| = 24$ . (Find the number of positions to which an adjacent pair of vertices can be sent and argue that fixing one edge uniquely determines the isometry.)

For reference, consider the following diagram of an octahedron.

Let $\theta$ be an orientation-preserving isometry of the octahedron; that is, if the vertices of a face, read clockwise from outside the figure, are $XYZ$ , then $\theta(X) \theta(Y) \theta(Z)$ are the vertices of the corresponding face, read clockwise from outside the figure, of the isometric copy.
There are 6 possibilities for $\theta{1}$ . Once $\theta(1)$ is chosen, there are 4 possibilities for $\theta(2)$ . Then the rest of $\theta$ is determined uniquely by orientation. Thus there are $6 \cdot 4 = 24$ possibilities for $\theta$ , add distinct.

Algebra Solutions

Find the number of orientation-preserving isometries of an octahedron

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