Induced class function

Definition
Suppose $$G$$ is a group and $$H$$ is a subgroup of finite index in $$G$$. Suppose $$f$$ is a class function on $$H$$ (i.e., a function on $$H$$ that is constant on each conjugacy class of $$H$$). Then, the induced class function on $$G$$, denoted $$\operatorname{Ind}_H^G(f)$$ is defined by the following summation over a left transversal $$S$$ of $$H$$ in $$G$$:

$$\! \operatorname{Ind}_H^G(f) = x \mapsto \sum_{s \in S} f_0(s^{-1}xs)$$

where $$f_0(g)$$ is $$f(g)$$ if $$g \in H$$ and $$0$$ otherwise.

Note that this is well-defined (independent of $$S$$_ precisely because $$f$$ is a class function on $$H$$, so replacing $$s$$ by $$sh, h \in H$$, gives an element conjugate via $$H$$ and hence the same value of $$f_0$$.

Facts

 * Character of induced representation equals induced class function from character: If $$f$$ is the character of a linear representation, then $$\operatorname{Ind}_H^G(f)$$ is the character of the induced representation from $$H$$ to $$G$$. it is often called the induced character from $$f$$.
 * Similarly, if $$f$$ is a virtual character, the induced class function is also a virtual character.