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
[hide]- 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 k∈N be given. The symmetric group on k symbols, denoted Sk, is the permutation group on {1,2,…,k−1,k}⊂N. The set of the group is the set of all permutations on {1,2,…,k−1,k}. See proof that the symmetric group is actually a group for details.
- Identity element: e:{1,…,k}→{1,…,k} which acts as so: e:i↦i - this is the identity permutation, it does nothing.
- The group operation is ordinary function composition, for σ,τ∈Sk we define:
- στ:=σ∘τ with: στ:{1,…,k}→{1,…,k} by στ:i↦σ(τ(i))
- Caveat:Be careful, a lot of authors (Allenby, McCoy to name 2) go left-to-right and write iσ for what we'd use σ(i) or σi at a push for. Then στ would be τ∘σ in our notation
- στ:=σ∘τ with: στ:{1,…,k}→{1,…,k} by στ:i↦σ(τ(i))
See also
References
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