Algebra Solutions: Compute the number of orientation-preserving isometries of an icosahedron

Let $G$ be the group of rigid motions (i.e., orientation preserving isometries) of an icosahedron in $\mathbb{R}^3$ . Show that $|G| = 60$ . (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 icosahedron.

Let $\theta$ be an orientation-preserving isometry of the icosahedron; 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 12 possibilities for $\theta(1)$ . Once $\theta(1)$ is chosen, there are 5 possibilities for $\theta(2)$ . Once these are chosen, the rest of $\theta$ is determined uniquely by orientation. Thus there are $12 \cdot 5 = 60$ possibilities for $\theta$ , all distinct. $\blacksquare$

Algebra Solutions

Compute the number of orientation-preserving isometries of an icosahedron

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