Cycle (permutation)

A cycle is a type of permutation.

Let be the symmetric group on a set . Let be an element of , and let be the subgroup of generated by . Then is a cycle if has exactly one orbit (under the operation of ) which does not consist of a single element. This orbit is called the support of , and is sometimes denoted $\text{supp}(\zeta})$ (Error compiling LaTeX. Unknown error_msg).

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See also

• Symmetric group
• Permutation
• Orbit