proof of Frobenius reciprocity

We prove the slightly more general result

Theorem 0.1.

If $G$ is a finite group with subgroup $H$, $\alpha$ a class function on $H$ and $\beta$ a class function on $G$, then

$${⟨\alpha {\uparrow }_H^G,\beta ⟩}_G={⟨\alpha ,\beta {\downarrow }_H^G⟩}_H$$

Here we use ${\uparrow }_H^G$ to refer to the induction (http://planetmath.org/InducedRepresentation) to $G$ of a class function on $H$, and ${\downarrow }_H^G$ to refer to the restriction (http://planetmath.org/RestrictionRepresentation) of a class function on $G$ to one on $H$.

Proof.

$${⟨\alpha {\uparrow }_H^G,\beta ⟩}_G=\frac{1}{\left|G\right|}\sum _{g\in G}\left(\frac{1}{\left|H\right|}\sum _{\begin{array}{c}t\in G\\ t^{-1}gt\in H\end{array}}\alpha (t^{-1}gt)\right)\overline{\beta (g)}=\frac{1}{\left|G\right|\left|H\right|}\sum _{t\in G}\left(\sum _{\begin{array}{c}g\in G\\ t^{-1}gt\in H\end{array}}\alpha (t^{-1}gt)\right)\overline{\beta (g)}$$

Since $\beta$ is a class function, this is the same as

$$\frac{1}{\left|G\right|\left|H\right|}\sum _{\begin{array}{c}t\in G\\ g\in G\\ t^{-1}gt\in H\end{array}}\alpha (t^{-1}gt)\overline{\beta (t^{-1}gt)}=\frac{1}{\left|G\right|\left|H\right|}\sum _{h\in H}\sum _{\begin{array}{c}t\in G\\ g\in G\\ t^{-1}gt=h\end{array}}\alpha (h)\overline{\beta (h)}$$

Clearly for every $h\in H,t\in G$ there is a unique $g\in G$ with $t^{-1}gt=h$, so every element of $H$ is counted $\left|G\right|$ times by the sum. Thus the sum is equal to

$$\frac{\left|G\right|}{\left|G\right|\left|H\right|}\sum _{h\in H}\alpha (h)\overline{\beta (h)}=\frac{1}{\left|H\right|}\sum _{h\in H}\alpha (h)\overline{\beta (h)}={⟨\alpha ,\beta {\downarrow }_H^G⟩}_H$$

∎