= Rotational symmetry group of a regular dodecahedron
{title2=$\operatorname{Rot}(D)\cong A_5$}
The <rotations in three dimensions> preserving a <dodecahedron> form a group of order $60$. There are $24$ nonidentity rotations about the six axes through opposite face centres, $20$ about the ten axes through opposite vertices, and $15$ half-turns about the fifteen axes through opposite edge midpoints. Its nonidentity <conjugacy classes> have sizes $12,12,20,15$: the <centralizer> of a rotation of order five or three consists of rotations about its oriented axis and has order five or three, respectively, while all edge half-turns are conjugate. A <normal subgroup> is a union of whole <conjugacy classes> containing the identity. No proper nontrivial sum of these class sizes plus one divides $60$, proving that it is a <simple group>. The <centralizers> of the fifteen <involutions> are five distinct <Klein four-groups>, each containing three <involutions>. Conjugation permutes these five subgroups nontrivially. The kernel is a <normal subgroup>, so simplicity makes this nontrivial action injective. Its sign map to a group of order two must be trivial, since simplicity would otherwise force an impossible injection of sixty elements into two. Thus its image lies in the <alternating group> $A_5$, and equality of orders makes the image all of $A_5$.
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