Editing Conjugacy class (section)

==Examples==
The symmetric group [[Dihedral group of order 6|<math>S_3,</math>]] consisting of the 6 [[permutation]]s of three elements, has three conjugacy classes:

# No change <math>(abc \to abc)</math>.  The single member has order 1.
# [[Cyclic permutation#Transpositions|Transposing]] two <math>(abc \to acb, abc \to bac, abc \to cba)</math>.  The 3 members all have order 2.
# A [[cyclic permutation]] of all three <math>(abc \to bca, abc \to cab)</math>.  The 2 members both have order 3.

These three classes also correspond to the classification of the [[Isometry group|isometries]] of an [[equilateral triangle]].

[[File:Symmetric group S4; conjugacy table.svg|thumb|300px|Table showing <math>bab^{-1}</math> for all pairs <math>(a, b)</math> with <math>a, b \in S_4</math> <small>(compare [[:File:Symmetric group 4; permutation list.svg|numbered list]])</small>. Each row contains all elements of the conjugacy class {{nowrap|of <math>a,</math>}} and each column contains all elements of <math>S_4.</math>]]
The '''symmetric group [[v:Symmetric group S4|<math>S_4,</math>]]''' consisting of the 24 permutations of four elements, has five conjugacy classes, listed with their description, [[Permutation#Cycle type|cycle type]], member order, and members:

# No change.  Cycle type = [1<sup>4</sup>].  Order = 1.  Members = { (1, 2, 3, 4) }.  The single row containing this conjugacy class is shown as a row of black circles in the adjacent table. 
# Interchanging two (other two remain unchanged).  Cycle type = [1<sup>2</sup>2<sup>1</sup>].  Order = 2.  Members = { (1, 2, 4, 3), (1, 4, 3, 2), (1, 3, 2, 4), (4, 2, 3, 1), (3, 2, 1, 4), (2, 1, 3, 4) }).  The 6 rows containing this conjugacy class are highlighted in green in the adjacent table.
# A cyclic permutation of three (other one remains unchanged).  Cycle type = [1<sup>1</sup>3<sup>1</sup>].  Order = 3.  Members = { (1, 3, 4, 2), (1, 4, 2, 3), (3, 2, 4, 1), (4, 2, 1, 3), (4, 1, 3, 2), (2, 4, 3, 1), (3, 1, 2, 4), (2, 3, 1, 4) }).  The 8 rows containing this conjugacy class are shown with normal print (no boldface or color highlighting) in the adjacent table.
# A cyclic permutation of all four.  Cycle type = [4<sup>1</sup>].  Order = 4.  Members = { (2, 3, 4, 1), (2, 4, 1, 3), (3, 1, 4, 2), (3, 4, 2, 1), (4, 1, 2, 3), (4, 3, 1, 2) }).  The 6 rows containing this conjugacy class are highlighted in orange in the adjacent table.
# Interchanging two, and also the other two.  Cycle type = [2<sup>2</sup>].  Order = 2.  Members = { (2, 1, 4, 3), (4, 3, 2, 1), (3, 4, 1, 2) }).  The 3 rows containing this conjugacy class are shown with boldface entries in the adjacent table.

The [[Octahedral symmetry#The isometries of the cube|proper rotations of the cube]], which can be characterized by permutations of the body diagonals, are also described by conjugation in <math>S_4.</math>

In general, the number of conjugacy classes in the '''symmetric group <math>S_n</math>''' is equal to the number of [[integer partition]]s of <math>n.</math>  This is because each conjugacy class corresponds to exactly one partition of <math>\{ 1, 2, \ldots, n \}</math> into [[Cycle notation|cycles]], up to permutation of the elements of <math>\{ 1, 2, \ldots, n \}.</math>

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

Example

Let G = <math>S_3</math>

<math>a = ( 2 3 )</math>

<math>x = ( 1 2 3 )</math>

<math>x^{-1} = ( 3 2 1 )</math>

Then <math>x a x^{-1}</math>

<math>= ( 1 2 3 ) ( 2 3 ) ( 3 2 1 ) = ( 3 1 )</math>

<math>= ( 3 1 )</math> is Conjugate of <math>( 2 3 )</math>