Symmetric groups are rational

This article describes a basic fact about permutations, or about the symmetric group or alternating group.
View a complete list of basic facts about permutations

Statement

The Symmetric group (?) on any set (finite or infinite) is a 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

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.