The group associated with the dodecahedron and the icosahedron has a structure radically different from all groups we have examined up to now. Galois, in the course of his investigation of the solvability of algebraic equations, discovered that the group of congruence motions of a regular icosahedron has many proper subgroups, but none of these is a normal subgroup. A group with no normal proper subgroups is called simple.

The dodecahedron and icosahedron have isomorphic groups of congruence motions since the two figures are dual figures (p. 142): the “centers” of the twelve regular pentagons forming the faces of a dodecahedron are the vertices of an icosahedron; and the “centers” of the twenty equilateral triangles forming the faces of an icosahedron are the vertices of a dodecahedron. The group of congruence motions of one figure is the “same” as the group of congruence motions of the other.

We shall now count the elements of the icosahedral group. If one vertex of an icosahedron is fixed in “apex” position, then a 72° counter-clockwise rotation of period 5 generates all congruence motions that leave the apex vertex fixed; see Figure 16.1. Since each of twelve vertices may be brought into apex position, the order of the icosahedral group is 60.

The order of A5 is ½5! = 60 (see p. 146), and, in fact, the icosahedral group is isomorphic to A5. The following is a sketch of a procedure the reader can follow to convince himself that this assertion is true.