Difference between revisions of "Symmetric group"
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*** {{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}} | *** {{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}} | ||
==See also== | ==See also== | ||
| − | * {{Cycle notation|(group theory)}} | + | * {{link|Cycle notation|(group theory)}} |
** [[Every element of the symmetric group can be written as the product of disjoint cycles]] | ** [[Every element of the symmetric group can be written as the product of disjoint cycles]] | ||
** {{link|Transposition|group theory}} - a {{M|2}}-cycle. | ** {{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]] | *** [[Every element of the symmetric group can be written as an even or odd number of transpositions, but not both]] | ||
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==References== | ==References== | ||
<references/> | <references/> | ||
{{Group theory navbox|plain}} | {{Group theory navbox|plain}} | ||
{{Definition|Group Theory|Abstract Algebra}} | {{Definition|Group Theory|Abstract Algebra}} | ||
Revision as of 11:39, 30 November 2016
Stub grade: A*
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Demote to grade D once fleshed out and referenced
Contents
- 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
- [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]
See also
References
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