Symmetric group

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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 [ilmath]k\in\mathbb{N} [/ilmath] be given. The symmetric group on [ilmath]k[/ilmath] symbols, denoted [ilmath]S_k[/ilmath], is the permutation group on [ilmath]\{1,2,\ldots,k-1,k\}\subset\mathbb{N} [/ilmath]. The set of the group is the set of all permutations on [ilmath]\{1,2,\ldots,k-1,k\} [/ilmath]. See proof that the symmetric group is actually a group for details.

• Identity element: [ilmath]e:\{1,\ldots,k\}\rightarrow\{1,\ldots,k\} [/ilmath] which acts as so: [ilmath]e:i\mapsto i[/ilmath] - this is the identity permutation, it does nothing.
• The group operation is ordinary function composition, for [ilmath]\sigma,\tau\in S_k[/ilmath] we define:
  • [ilmath]\sigma\tau:\eq \sigma\circ\tau[/ilmath] with: [ilmath]\sigma\tau:\{1,\ldots,k\}\rightarrow\{1,\ldots,k\} [/ilmath] by [ilmath]\sigma\tau:i\mapsto\sigma(\tau(i))[/ilmath]
    • Caveat:Be careful, a lot of authors (Allenby, McCoy to name 2) go left-to-right and write [ilmath]i\sigma[/ilmath] for what we'd use [ilmath]\sigma(i)[/ilmath] or [ilmath]\sigma i[/ilmath] at a push for. Then [ilmath]\sigma\tau[/ilmath] would be [ilmath]\tau\circ\sigma[/ilmath] in our notation

See also

• Template:Cycle notation
  • Every element of the symmetric group can be written as the product of disjoint cycles
  • Transposition - a [ilmath]2[/ilmath]-cycle.
    • Every element of the symmetric group can be written as an even or odd number of transpositions, but not both