Editing Symmetric group (section)

=== Cycles ===
A [[cyclic permutation|cycle]] of ''length'' ''k'' is a permutation ''f'' for which there exists an element ''x'' in {1, ..., ''n''} such that ''x'', ''f''(''x''), ''f''<sup>2</sup>(''x''), ..., ''f''<sup>''k''</sup>(''x'') = ''x'' are the only elements moved by ''f''; it conventionally is required that {{nowrap|''k'' ≥ 2}} since with {{nowrap|1=''k'' = 1}} the element ''x'' itself would not be moved either. The permutation ''h'' defined by
: <math>h = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 1 & 3 & 5\end{pmatrix}</math>
is a cycle of length three, since {{nowrap|1=''h''(1) = 4}}, {{nowrap|1=''h''(4) = 3}} and {{nowrap|1=''h''(3) = 1}}, leaving 2 and 5 untouched. We denote such a cycle by {{nowrap|(1 4 3)}}, but it could equally well be written {{nowrap|(4 3 1)}} or {{nowrap|(3 1 4)}} by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are ''disjoint'' if they have disjoint subsets of elements. Disjoint cycles [[Commutative property|commute]]: for example, in S<sub>6</sub> there is the equality {{nowrap|1=(4 1 3)(2 5 6) = (2 5 6)(4 1 3)}}. Every element of S<sub>''n''</sub> can be written as a product of disjoint cycles; this representation is unique [[up to]] the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point.

Cycles admit the following conjugation property with any permutation <math>\sigma</math>, this property is often used to obtain its [[Symmetric group#Generators and relations|generators and relations]].

:<math>\sigma\begin{pmatrix} a & b & c & \ldots \end{pmatrix}\sigma^{-1}=\begin{pmatrix}\sigma(a) & \sigma(b) & \sigma(c) & \ldots\end{pmatrix}</math>