# Orbit of Permutations

Let be a permutation on a set . If a relation is defined on such that

for some integrals , we observe that the relation is:

**(i) Reflexive:** the relation is reflexive, i.e. , now we can define the reflexive property according to the above definition, because

**(ii) Symmetric:** the relation is symmetric, i.e. , we can show this relation by using the definition of orbit of permutation, because

for some integers

**(iii) Transitive:** the above relation is transitive, i.e. and implies , now we can prove this transitive property by using the above definition of orbit of permutation, because

for some integers and

for some integer

Thus the above defined relation is an equivalence relation on and hence we partition it into mutually disjoint classes. Each equivalence class determined by the relation is called an orbit of .