We need to prove that the symmetric group $S_n$ acts transitively in it's usual action as permutations on $ A = \{ 1,2,3,….,n \}$.
We know that the action of a group $G$ is called transitive if there is only one orbit(Number of equivalence classes of an element).
Also , index of a stabilizer of an element gives the number of elements in the equivalence class of that element.
So , my approach is , if I somehow show that $G_i$ which is the stabilizer of any point $i$ in $A$ has index $n$ then our job is done.
$G_i = s \in S_n | s \cdot i = i $ ,
But how to show that it has index $n$ ?
Or how to show it has $n$ number of cosets ? Could anyone help ?
Best Answer
Hint: Notice that the cycle $(k,l)\in S_n$ for any $k,l\in \{1,2,..,n\}$.