Symmetric groups are rational

Statement
The fact about::symmetric group on any set (finite or infinite) is a fact about::rational group.

Related facts

 * Finitary symmetric group on infinite set is rational
 * Finitary alternating group on infinite set is rational
 * Symmetric groups are ambivalent
 * Symmetric groups are strongly ambivalent

Facts used

 * 1) Cycle decomposition theorem
 * 2) Cycle type determines conjugacy class

Proof outline

 * 1) Take any permutation $$g$$. Express it using its cycle decomposition.
 * 2) Show that any other permutation $$h$$ generating the same cyclic group has the same cycle type as $$g$$.
 * 3) Use the fact that any two permutations with the same cycle type, are conjugate inside the symmetric group.