# Orbit of Permutations

Let $f$ be a permutation on a set $S$. If a relation $\sim$ is defined on $S$ such that
$a \sim b \Leftrightarrow {f^{\left( n \right)}}\left( a \right) = b$

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
$a \sim a \Leftrightarrow {f^{\left( n \right)}}\left( a \right)$
$= I\left( a \right) = a,\forall a \in S$

(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
$a \sim a \Leftrightarrow {f^{\left( n \right)}}\left( a \right) = b$
for some integers $n$
$\Rightarrow a = {f^{ – \left( n \right)}}\left( b \right)$
$\Rightarrow b \sim a,\,\forall a,b \in S$

(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
$\Rightarrow {f^{\left( n \right)}}\left( a \right) = b,\,\,\,{f^{\left( m \right)}}\left( b \right) = c$ for some integers $n$ and $m$
$\Rightarrow {f^m}\left( {{f^n}\left( a \right)} \right) = {f^m}\left( b \right) = c$
$\Rightarrow {f^{m + n}}\left( a \right) = c$  for some integer $m + n$
$\Rightarrow a \sim c$

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 $f$.