Icosahedral symmetry as permutation group Hopefully an easy question: the icosahedral group of order 60 (orientation preserving symmetries of a regular icosahedron) is isomorphic to the alternating group on 5 points.  In terms of the icosahedron, what are the 5 "points"?
It would be ideal if the "points" were actually points.  The wikipedia article mentions some compounds of inscribed solids, and I think I'd need a physical demonstration to see that.  However, according to this question and this question, we should be able to give just a few points, so that the stabilizer of those points has order 12.
However, I am not sure what the points would be.  Maybe they are the vertices of an associated tetrahedron.  It would be nice if it was easy to describe those vertices, as I certainly don't see any tetrahedra myself, but I can imagine a collection of 4 vertices easily enough.
 A: 
The icosahedral group $I$ permutes the 5 colors in this coloring of the icosahedron. A rotation of the icosahedron about a vertex gives a 5 cycle permutation of the colors. A rotation of the icosahedron about a triangle face is a 3 cycle permutation of colors. A rotation about an edge gives (ab)(cd) signature.
As @draks says I think the action is more obvious that this coloring is preserved if you think of inscribed tetrahedra in the dodecahedron, but this is the icosahedron equivalent. The corners of those tetrahedra partition the vertices of the dodecahedron, corresponding to this partition of faces in the icosahedron.
Proof $I$ acts on the coloring as $A_5$:
Up to reflection this coloring is unique in having the properties that: each color has 4 faces, no pair of faces sharing an edge or vertex are the same color, and every vertex is incident to all 5 colors. Clearly those properties are preserved by $I$. I don't have a good uniqueness argument other than after you assign the colors around a vertex, you have a single choice you can make before all other colors are forced by the non-adjacency conditions. Therefore $I$ must preserve this coloring.
Once you convince yourself that this coloring is preserved, then the action of the icosahedral group on the set of colors must be $A_5$, since $A_5$ is simple. Any kernel of the action must be trivial.
