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