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# Permutations

Definition 5.6.1   A permutation of a set is a bijection .

For the set , the notation

denotes the permutation of such that , , etc.

Definition 5.6.2   Let be a set and be a subset of . Let be a permutation of verifying the following properties:
(i)
, , ..., (renumbering the elements of if necessary;
(ii)
for any , .
Then is called a cycle.

For example, in is a cycle, whose notation can be shortened to .

Proposition 5.6.3   Any permutation is the product of disjoint cycles.

Subsections

root 2002-06-10