Exercise 1.2.13 (Group of rigid motions of the icosahedron)

Proof. Let $ℐ$ a regular icosahedron ( $20$ faces, $12$ vertices), and $G$ be the group of rigid motions of $ℐ$.

$G$ acts on the set $ℐ$ by $g\cdot M=g(M)$ $(g\in G,M\in 𝒟)$. Using rotations of angles $2\pi ∕3$ and axe $0I$, where $I$ is a centre of a face, we can send $A$ on any vertex. So the action is transitive, there is a unique orbit ${\mathcal{}O}_A$, and $|{\mathcal{}O}_A|=|ℐ|=12$.

The stabilizer $G_A$ of $A$ has five elements: $G_A=\{e,\rho ,{\rho }^2,{\rho }^3,{\rho }^4\}$ where $\rho$ is the rotation of angle $2\pi ∕5$ and axe $\mathit{OA}$. By the fundamental theorem of group actions, $(G:G_A)=|\mathcal{}{O}_A|$, so

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