Exercise 4.5.43 (Group of rigid motions of a dodecahedron)

Proof.

Let $G^{\prime }$ be the group of rigid motions in ${ℝ}^3$ of a dodecahedron. By Exercise 1.2.12, $|G^{\prime }|=60$.

The rotations of angles $k\frac{2\pi }{5}(0\le k\le 4)$ around the axes joining the two centers of opposite faces form $6$ Sylow $5$-subgroups of $G^{\prime }$. The argument given in Exercise 42 proves that

(Since the icosahedron and the dodecahedron are dual solids, the two groups $G$ and $G^{\prime }$ are identical : $G^{\prime }=G\simeq A_5$.) □