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: '''Note: ''' the ''symmetric group'' is a ''[[permutation group]]'' on finitely many symbols, see [[permutation group]] (which uses the same notation) for the more general case.
==Definition==
Let {{M|k\in\mathbb{N} }} be given. The ''symmetric group'' on {{M|k}} symbols, denoted {{M|S_k}}, is the [[permutation group]] on {{M|\{1,2,\ldots,k-1,k\}\subset\mathbb{N} }}. The [[set]] of the [[group]] is the set of all [[Permutation of a set|permutations]] on {{M|\{1,2,\ldots,k-1,k\} }}. See [[proof that the symmetric group is actually a group]] for details.
* Identity element: {{M|e:\{1,\ldots,k\}\rightarrow\{1,\ldots,k\} }} which acts as so: {{M|e:i\mapsto i}} - this is the identity permutation, it does nothing.
* The group operation is ordinary function composition, for {{M|\sigma,\tau\in S_k}} we define:
** {{M|\sigma\tau:\eq \sigma\circ\tau}} with: {{M|\sigma\tau:\{1,\ldots,k\}\rightarrow\{1,\ldots,k\} }} by {{M|\sigma\tau:i\mapsto\sigma(\tau(i))}}
*** {{Caveat|Be careful, a lot of authors (Allenby, McCoy to name 2) go left-to-right and write {{M|i\sigma}} for what we'd use {{M|\sigma(i)}} or {{M|\sigma i}} at a push for. Then {{M|\sigma\tau}} would be {{M|\tau\circ\sigma}} in our notation}}
==Permutation notation==
The "base" way to write a permutation, {{M|\sigma\in S_k}}, is as a table:
* {{MM|1=\left(\begin{array}{ccccc}1 & 2 & \cdots & k-1 & k\\ \sigma(1) & \sigma(2) & \cdots & \sigma(k-1) & \sigma(k)\end{array}\right)}}
The top row is an element of the domain of {{M|\sigma}} considered as a [[function]] and thing below it is the image of that element under {{M|\sigma}}


This notation quickly becomes heavy so we switch to {{link|cycle notation|group theory}}, which we demonstrate below.

Note, however, that in order to use cycle notation we require the following:
* [[Every element of the symmetric group, if not a cycle itself, can be expressed as a product of disjoint cycles]] 
===[[Example:Permutation (group theory) of S5|Example]]===
{{:Example:Permutation (group theory) of S5/Body}}
See [[Cycle notation (group theory)]] for more information.
==See also==
* {{link|Cycle notation|(group theory)}}
** [[Every element of the symmetric group, if not a cycle itself, can be expressed as a product of disjoint cycles]] 
** {{link|Transposition|group theory}} - a {{M|2}}-cycle.
*** [[Every element of the symmetric group can be written as an even or odd number of transpositions, but not both]]

==References==
<references/>
{{Group theory navbox|plain}}
{{Definition|Group Theory|Abstract Algebra}}