Splitting implies characters separate conjugacy classes

Statement
For a finite group $$G$$, any fact about::splitting field $$k$$ has the following property: given any two distinct fact about::conjugacy classes of $$G$$, there is a finite-dimensional representation $$\varphi$$ of $$G$$ over $$k$$ such that the character of $$\varphi$$ takes different values on the two conjugacy classes.

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, in fact: characters span class functions not implies splitting (this is due to the Schur index phenomenon).

On the other hand, see characters span class functions iff they separate conjugacy classes iff field contains field generated by character values for alternative characterizations of a field where the characters separate the conjugacy classes.

Facts used

 * 1) uses::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.