[[Group theory MOC]]
# Class function

A **class function** $f \in X[G] : G \to X$ from a group $G$ to some set $X$ is a function constant for each [[Conjugation by an element|conjugacy class]], #m/def/group
i.e. for all $x,y \in G$
$$
\begin{align*}
f(yxy^{-1}) = f(x)
\end{align*}
$$

## Examples

- [[Group character]]

## Properties

- If $X$ is a ring, we consider functions as elements of the [[Group ring]]
  - [[Centre of the group ring]]
- [[Tensor powers of a faithful representation contain all irreps]]

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#state/tidy  | #lang/en | #SemBr