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In mathematics, the cycles of a permutation π of a finite set S correspond bijectively to the orbits of the subgroup generated by π acting on S. These orbits are subsets of S that can be written as { c1,..., cl }, such that
The corresponding cycle of π is written as ; this expression is not unique since c1 can be chosen to be any element of the orbit.
The size l of the orbit is called the length of the corresponding cycle; when l = 1, the single element in the orbit is called a fixed point of the permutation.
A permutation is determined by giving an expression for each of its cycles, and one notation for permutations consist of writing such expressions one after another in some order. For example, let
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