# Orbit of Permutations

Let $f$ be a permutation on a set $S$. If a relation $\sim$ is defined on $S$ such that

for some integrals $n\forall \,a,b \in S$, we observe that the relation is:

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

(ii) Symmetric: the relation is symmetric, i.e. $a \sim b \Rightarrow b \sim a$, we can show this relation by using the definition of orbit of permutation, because

for some integers $n$

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

for some integers $n$ and $m$

for some integer $m + n$

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