# Symmetric groups are rational

From Groupprops

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

## Proof

### Proof outline

- Take any permutation . Express it using its cycle decomposition.
- Show that any other permutation generating the same cyclic group has the same cycle type as .
- Use the fact that any two permutations with the same cycle type, are conjugate inside the symmetric group.