Editing Symmetric group (section)

== Conjugacy classes ==

The [[conjugacy class]]es of S<sub>''n''</sub> correspond to the [[Permutation#Cycle type|cycle types]] of permutations; that is, two elements of S<sub>''n''</sub> are conjugate in S<sub>''n''</sub> if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S<sub>5</sub>, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of S<sub>''n''</sub> can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another.  Continuing the previous example,
<math display="block">k = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 4 & 3 & 2 & 5\end{pmatrix},</math>
which can be written as the product of cycles as (2 4).
This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is,
<math display="block">(2~4)\circ(1~2~3)(4~5)\circ(2~4)=(1~4~3)(2~5).</math>
It is clear that such a permutation is not unique.

Conjugacy classes of S<sub>''n''</sub> correspond to [[integer partition]]s of ''n'': to the partition {{nowrap|1=''μ'' = (''μ''<sub>1</sub>, ''μ''<sub>2</sub>, ..., ''μ''<sub>''k''</sub>)}} with <math display="inline">n=\sum_{i=1}^k \mu_i</math> and {{nowrap|''μ''<sub>1</sub> ≥ ''μ''<sub>2</sub> ≥ ... ≥ ''μ''<sub>''k''</sub>}}, is associated the set ''C''<sub>''μ''</sub> of permutations with cycles of lengths {{nowrap|''μ''<sub>1</sub>, ''μ''<sub>2</sub>, ..., ''μ''<sub>''k''</sub>}}. Then ''C''<sub>''μ''</sub>  is a conjugacy class of S<sub>''n''</sub>, whose elements are said to be of cycle-type <math>\mu</math>.