Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 4 / 11G (Groups, Rings and Modules)

First, the orbit-stabilizer theorem applied to a face gives $|G|=20\cdot 3=60$: its stabilizer subgroup consists of the identity and the two rotations through $120^{\circ }$ about the axis through the centres of that face and its opposite. A rotation preserving a face must preserve its outward normal and cyclically permute its three vertices, so there are no further stabilizing rotations.

The conjugacy classes are as follows:

There are $30/2=15$ edge axes, $20/2=10$ face axes, and $12/2=6$ vertex axes. They give respectively $15$, $2\cdot 10=20$, and $4\cdot 6=24$ distinct nonidentity rotations, already accounting for all $59$ nonidentity elements.

To justify the class assertions, conjugation by a rotation carries the rotation axis to its image and preserves the rotation angle about the transported oriented axis. The transitive group action on edges puts all half-turns in one conjugacy class. The transitive group action on faces puts the $120^{\circ }$ rotations defined using outward face normals in one conjugacy class; opposite faces encode opposite signs about the same unoriented axis. The same reasoning with outward vertex directions gives a class of twelve $72^{\circ }$ rotations and a class of twelve $144^{\circ }$ rotations, including their inverses. These last classes cannot merge: the matrix trace of a three-dimensional rotation is $1+2\cos \theta$, which differs for $\theta =2\pi /5$ and $4\pi /5$. Thus the list is complete.

The face action gives $|G|=60$, as calculated above. Every normal subgroup contains the identity and is a union of whole conjugacy classes. Thus its possible order is $1$ plus a subset sum of $15,20,12,12$. The distinct resulting numbers areBy Lagrange's theorem for finite groups, a subgroup order divides $60$. Only $1$ and $60$ in this list do so. Consequently the normal subgroups are $\{1\}$ and $G$, and the rotational symmetry group of a regular icosahedron is a simple group.