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 a finite set is a Rational group (?), i.e., it satisfies the following equivalent conditions:

1. If $r$ is relatively prime to the order of the group, then for any $g$ in the group, $g$ and $g^r$ are conjugate.
2. Every character of the group is rational-valued.
3. Every character of the group is integer-valued.

Definitions used

Symmetric group

Further information: symmetric group

Rational group

Further information: rational group

Related facts

• Finitary symmetric group on infinite set is rational
• Finitary alternating group on infinite set is rational
• Symmetric group on infinite set is rational

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 if $r$ is relatively prime to the order of the group, then $g$ and $g^r$ have the same cycle type.
3. Use the fact that any two permutations with the same cycle type, are conjugate inside the symmetric group.