Question on class functions

Question

Let $G$ be a group and $f$ be a class function on the group.

We know that if $g,h$ are in the same conjugacy class, then $f(g) = f(h)$.

However, is it true that if $f(g) = f(h)$, then $g$ and $h$ are in the same conjugacy class?

Answer 1

The answer is no, as noted in the comments. For example if $f$ is constant, then $f(g) = f(h)$ for all $g, h \in G$.

However, if $\varphi(g) = \varphi(h)$ for all class functions $\varphi$ on $G$ (not just $\varphi = f$), then $g$ and $h$ are conjugate in $G$.

Here is the proof: let $C$ be the conjugacy class of $g$ in $G$. Define $\varphi: G \rightarrow \mathbb{C}$ by $$\varphi(x) = \begin{cases} 1, & \text{ if } x \in C. \\ 0, & \text{ if }x \not\in C.\end{cases}$$ Then $\varphi$ is a class function, so $\varphi(g) = \varphi(h)$ implies that $g$ and $h$ are conjugate in $G$.