Frobenius reciprocity

Plain statement
Let $$H \le G$$ be finite groups, $$k$$ be a field whose characteristic does not divide the order of $$G$$, and $$f_1,f_2$$ be class functions on $$H$$ and $$G$$ respectively taking values in $$k$$. In other words, $$f_1:H \to k$$ and $$f_2:G \to k$$ are functions with the property that $$f_1$$ is constant on conjugacy classes in $$H$$ and $$f_2$$ is constant on conjugacy classes in $$G$$.

Then, the inner product of $$\operatorname{Ind}(f_1)$$ with $$f_2$$ (in $$G$$) equals the inner product of $$f_1$$ and $$\operatorname{Res}(f_2)$$ (in $$H$$).

In symbols:

$$\langle \operatorname{Ind}(f_1)_H^G,f_2 \rangle_G = \langle f_1, \operatorname{Res}(f_2)_H^G\rangle_H$$

This also applies in particular to the case when $$f_1,f_2$$ are the characters of linear representations.

Category-theoretic statement
Let $$H \le G$$.

Consider the category of representations of $$G$$ (viz $$G$$-modules) and the category of representations of $$H$$, with a homomorphism of objects within a category, a vector space homomorphism that commutes with the $$G$$-action. Then, induction defines a functor from the category of $$H$$-modules to the category of $$G$$-modules, and restriction defines a functor from the category of $$G$$-modules to the category of $$H$$-modules. Frobenius reciprocity says that these two functors are adjoint functors.

Applications to relation between representation theory of group and subgroup

 * Maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroup
 * Maximum degree of irreducible representation of subgroup is less than or equal to maximum degree of irreducible representation of whole group