Permutation of a set
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Revision as of 23:59, 21 July 2016 by Alec (Talk | contribs) (Saving work, could really use a load on notation. However I'm familiar with that so notations are low priority.)
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Very important to get some work on the symmetric group in play, then this may be demoted. Demote to grade A once the notation section has been added, there's a lot to say there.
- Note: permutation on a set redirects here.
Contents
Definition
Let [ilmath]X[/ilmath] be any non-empty set, [ilmath]X[/ilmath]. A permutation on [ilmath]X[/ilmath][1][2] is:
- A bijective function, [ilmath]f:X\rightarrow X[/ilmath]. Recall that bijective means injective (1:1) and surjective (onto).
Claims:
- The collection of all permutations of a set forms a group under function composition - see the permutation group. The symmetric group is a special case of the permutation group when the set is finite.
References
- ↑ Rings, Fields and Groups - An introduction to abstract algebra - R. B. J. T. Allenby
- ↑ Abstract Algebra - Pierre Antoine Grillet
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