# Induced class function

From Groupprops

## Definition

Suppose is a group and is a subgroup of finite index in . Suppose is a class function on (i.e., a function on that is constant on each conjugacy class of ). Then, the **induced class function** on , denoted is defined by the following summation over a left transversal of in :

where is if and otherwise.

Note that this is well-defined (independent of _ precisely because is a class function on , so replacing by , gives an element conjugate via and hence the same value of .

## Facts

- Character of induced representation equals induced class function from character: If is the character of a linear representation, then is the character of the induced representation from to . it is often called the
*induced character*from . - Similarly, if is a virtual character, the induced class function is also a virtual character.