Permutation of a set
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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 nonempty 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
