Examples
The symmetric group S3, consisting of all 6 permutations of three elements, has three conjugacy classes:
- no change (abc → abc)
- interchanging two (abc → acb, abc → bac, abc → cba)
- a cyclic permutation of all three (abc → bca, abc → cab)
The symmetric group S4, consisting of all 24 permutations of four elements, has five conjugacy classes, listed with their cycle structures and orders:
- (1)4: no change (1 element)
- (2): interchanging two (6 elements)
- (3): a cyclic permutation of three (8 elements)
- (4): a cyclic permutation of all four (6 elements)
- (2)(2): interchanging two, and also the other two (3 elements)
In general, the number of conjugacy classes in the symmetric group Sn is equal to the number of integer partitions of n. This is because each conjugacy class corresponds to exactly one partition of {1, 2, ..., n} into cycles, up to permutation of the elements of {1, 2, ..., n}.
See also the proper rotations of the cube, which can be characterized by permutations of the body diagonals.
Read more about this topic: Conjugacy Class
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