Editing Alternating group (section)

== Example A<sub>5</sub> as a subgroup of 3-space rotations ==
[[File:A5_in_SO(3).gif|thumb|A<sub>5</sub> < SO<sub>3</sub>('''R''')
{{legend|gray|[[ball (mathematics)|ball]] – radius {{pi}} – [[principal homogeneous space]] of SO(3)}}
 {{legend|yellow|[[icosidodecahedron]] – radius {{pi}} – conjugacy class of 2-2-cycles}}
 {{legend|purple|[[icosahedron]] – radius 4{{pi}}/5 – half of the [https://groupprops.subwiki.org/wiki/Splitting_criterion_for_conjugacy_classes_in_the_alternating_group split] conjugacy class of 5-cycles}}
 {{legend|green|[[dodecahedron]] – radius 2{{pi}}/3 – conjugacy class of 3-cycles}}
 {{legend|red|icosahedron – radius 2{{pi}}/5 – second half of split 5-cycles}}
]]
[[File:Compound of five tetrahedra.png|thumb|Compound of five tetrahedra. A<sub>5</sub> acts on the dodecahedron by permuting the 5 inscribed tetrahedra. Even permutations of these tetrahedra are exactly the symmetric rotations of the dodecahedron and characterizes the {{nowrap|A<sub>5</sub> < SO<sub>3</sub>('''R''')}} correspondence.]]

A<sub>5</sub> is the group of isometries of a dodecahedron in 3-space, so there is a representation {{nowrap|A<sub>5</sub> → SO<sub>3</sub>('''R''')}}.

In this picture the vertices of the polyhedra represent the elements of the group, with the center of the sphere representing the identity element. Each vertex represents a rotation about the axis pointing from the center to that vertex, by an angle equal to the distance from the origin, in radians. Vertices in the same polyhedron are in the same conjugacy class. Since the conjugacy class equation for A<sub>5</sub> is {{nowrap|1=1 + 12 + 12 + 15 + 20 = 60}}, we obtain four distinct (nontrivial) polyhedra.

The vertices of each polyhedron are in bijective correspondence with the elements of its conjugacy class, with the exception of the conjugacy class of (2,2)-cycles, which is represented by an icosidodecahedron on the outer surface, with its antipodal vertices identified with each other. The reason for this redundancy is that the corresponding rotations are by {{pi}} radians, and so can be represented by a vector of length {{pi}} in either of two directions. Thus the class of (2,2)-cycles contains 15 elements, while the icosidodecahedron has 30 vertices.

The two conjugacy classes of twelve 5-cycles in A<sub>5</sub> are represented by two icosahedra, of radii 2{{pi}}/5 and 4{{pi}}/5, respectively. The nontrivial outer automorphism in {{nowrap|Out(A<sub>5</sub>) ≃ Z<sub>2</sub>}} interchanges these two classes and the corresponding icosahedra.