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 .
Some properties of cycles
Lemma. Let be a family of cycles of with pairwise disjoint supports . Then the commute. The product is then well defined as , for , and , for . Let be the subgroup generated by . Then the function is a bijection from to the orbits of containing more than one element.
Proof. Suppose and are of the . Then so by symmetry . This proves that the commute and justifies the definition of .
Suppose is a an orbit of with more than one element, and suppose . Then by our characterization of , must belong to , for some ; since is the orbit of , it follows that . Thus the mapping is a surjection from to the orbits of with more than one element; since it is evidently injective, it follows that this mapping is a bijection.
Theorem (cycle notation). Let be an element of . Then there exists a unique set of cycles of with pairwise disjoint supports such that
Proof. Let be the subgroup of generated by . Let be the family of nonempty orbits of . For all , let be the restriction of to ; let . Then by the lemma, Since the mapping must be a bijection from to the orbits of , it follows from the lemma that is unique.