# Class function

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## Definition

### Symbol-free definition

A class function' on a group is defined as a function (to any set) that takes the same value on any two conjugate elements. Equivalently, it is a function on the group that is constant on conjugacy classes, and hence descends to a function from the set of conjugacy classes.

### Definition with symbols

A class function on a group $G$ is a function $f$ from $G$ to some set $X$ such that $f(g) = f(hgh^{-1})$ for any $g,h \in G$.

## Particular cases

### Conjugacy classes of images are class functions

Let $\rho:G \to H$ be a homomorphism. Then the function that sends each $g$ to the conjugacy class of $\rho(g)$ is a class function. This follows from the fact that if two elements in $G$ are conjugate, their images in $H$ are also conjugate.

### Characters of linear representations are class functions

Further information: Character

For any linear representation, the character of that linear representation, viz the map that sends each group element to the trace of the corresponding linear operator, is a class function. This follows from the fact that the character depends only on the conjugacy class of the linear operator corresponding to the group element.