# Class function

## Contents

## 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 is a function from to some set such that for any .

## Particular cases

### Conjugacy classes of images are class functions

Let be a homomorphism. Then the function that sends each to the conjugacy class of is a class function. This follows from the fact that if two elements in are conjugate, their images in 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.