Splitting implies characters separate conjugacy classes

Statement

For a finite group $G$, any Splitting field (?) $k$is a Character-separating field (?): given any two distinct conjugacy classes, there is a finite-dimensional representation $\phi$ of $G$ over $k$ such that the character of $\phi$ takes different values on the two conjugacy classes.

Related facts

Corollaries

• Splitting implies class-separating: Given any two distinct conjugacy classes, we can find a finite-dimensional representation where the images are not conjugate as linear transformations.

Converse

The converse is not true, i.e., character-separating not implies splitting. In fact, characters span class functions not implies splitting (this is due to the Schur index phenomenon).

Facts used

1. Splitting implies characters span class functions

Proof

The proof follows directly from Fact (1). If it were true that there were two distinct conjugacy classes on which all characters were equal, then the span of the space of characters would not include any class function taking different values on the two conjugacy classes, and in particular would not include the indicator function for any one of the conjugacy classes. Hence, the span would not be all class functions.