Symmetric groups are rational-representation

This article gives the statement, and possibly proof, of a particular group or type of group (namely, Symmetric group (?)) satisfying a particular group property (namely, Rational-representation group (?)).

Statement

Let $n$ be a natural number. The symmetric group of degree $n$ is a Rational-representation group (?): all its representations over the complex numbers can be realized over the field of rational numbers.

Related facts

Weaker facts

• Symmetric groups are rational
• Symmetric groups are ambivalent

Related facts about alternating groups

• Classification of ambivalent alternating groups
• Finitary alternating group on infinite set is ambivalent

General information pages

• Linear representation theory of symmetric groups
• Linear representation theory of alternating groups