# Difference between revisions of "Characters span class functions iff they separate conjugacy classes iff field contains field generated by character values"

## Statement

Suppose $G$ is a group and $k$ is a field whose characteristic does not divide the order of $G$. The following are equivalent:

1. The characters of irreducible representations of $G$ over $k$ form a basis for the space of class functions.
2. The characters of irreducible representations of $G$ over $k$ span the space of class functions.
3. The characters of all finite-dimensional linear representations of $G$ over $k$ span the space of class functions.
4. Given any two distinct conjugacy classes of $G$, there is an irreducible representation whose character value is different on the two conjugacy classes.
5. $k$ contains a subfield $K$ that is isomorphic to the field generated by character values in its characteristic, i.e., there is a splitting field containing $k$ such that all irreducible representations of $G$ over the splitting field have character values in $k$.

Note that $k$ need not itself be a splitting field for $G$. For instance, the field of rational numbers satisfies these equivalent conditions for the quaternion group (see linear representation theory of quaternion group, faithful irreducible representation of quaternion group).

## Facts used

1. Splitting implies characters form a basis for space of class functions, splitting implies characters separate conjugacy classes