Induced class function

From Groupprops
Jump to: navigation, search

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