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Examples

The symmetric group S 3, {\displaystyle S_{3},} consisting of the 6 permutations of three elements, has three conjugacy classes:

These three classes also correspond to the classification of the isometries of an equilateral triangle.

The symmetric group S 4, {\displaystyle S_{4},} consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their description, cycle type, member order, and members:

The proper rotations of the cube, which can be characterized by permutations of the body diagonals, are also described by conjugation in S 4. {\displaystyle S_{4}.}

In general, the number of conjugacy classes in the symmetric group S n {\displaystyle S_{n}} is equal to the number of integer partitions of n. {\displaystyle n.} This is because each conjugacy class corresponds to exactly one partition of { 1, 2, …, n } {\displaystyle \{1,2,\ldots,n\}} into cycles, up to permutation of the elements of { 1, 2, …, n }. {\displaystyle \{1,2,\ldots,n\}.}

In general, the Euclidean group can be studied by conjugation of isometries in Euclidean space.

Example

Let G = S 3 {\displaystyle S_{3}}

a = (2 3)

x = (1 2 3)

x = (3 2 1)

Then xax

= (1 2 3) (2 3) (3 2 1) = (3 1)

= (3 1) is Conjugate of (2 3)