Definition
A permutation of a set X, which is a bijective function, is called a cycle if the action on X of the subgroup generated by has exactly one orbit with more than a single element. This notion is most commonly used when X is a finite set; then of course the orbit S in question is also finite. Let be any element of S, and put for any . Since by assumption S has more than one element, ; if S is finite, there is a minimal number for which . Then, and is the permutation defined by
and for any element of . The elements not fixed by can be pictured as
- .
A cycle can be written using the compact cycle notation (there are no commas between elements in this notation, to avoid confusion with a k-tuple). The length of a cycle, is the number of elements of its orbit of non-fixed elements. A cycle of length k is also called a k-cycle.
Read more about this topic: Cycle (mathematics)
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