Exercise 4.5.42 (Group of rigid motions of a icosahedron)

Question

Prove that the group of rigid motions in $\mathbb{R}^3$ of a icosahedron is isomorphic to $A_5$. [Recall that the order of this group is $60$: Exercise 12, Section 1.2.]

Richard Ganaye · Accepted Answer

\begin{proof} 
We assume the geometric properties of the icosahedron are known. We call diagonals the axes joining two opposite vertices: there are 6 such diagonals.

Let $G$ be the group of rigid motions in $\mathbb{R}^3$ of a icosahedron: the order of $G$ is $60$: cf. Exercise 12, Section 1.2.

 The rotations of angles $k \frac{2\pi}{5}\ (0 \leq k \leq 4)$ around such a diagonal form a Sylow $5$-subgroup of $G$, so there are $6$  Sylow $5$-subgroups. By Proposition 21, $G$ is simple, and by Proposition 23,
\begin{align*}
G \simeq A_5.
\end{align*}
\end{proof}
Note: This is the method chosen by Felix Klein to prove $G \simeq A_5$ (see [Felix Klein] Lectures on the Icosahedron, I §8.)

Exercise 4.5.42 (Group of rigid motions of a icosahedron)

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Exercise 4.5.42 (Group of rigid motions of a icosahedron)

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