# 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

 $\langle\alpha\negmedspace\uparrow_{H}^{G},\beta\rangle_{G}=\langle\alpha,\beta% \negmedspace\downarrow_{H}^{G}\rangle_{H}$

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

###### Proof.
 $\langle\alpha\negmedspace\uparrow_{H}^{G},\beta\rangle_{G}=\frac{1}{\left% \lvert G\right\rvert}\sum_{g\in G}\left(\frac{1}{\left\lvert H\right\rvert}% \sum_{\begin{subarray}{c}t\in G\\ t^{-1}gt\in H\end{subarray}}\alpha(t^{-1}gt)\right)\overline{\beta(g)}=\frac{1% }{\left\lvert G\right\rvert\left\lvert H\right\rvert}\sum_{t\in G}\left(\sum_{% \begin{subarray}{c}g\in G\\ t^{-1}gt\in H\end{subarray}}\alpha(t^{-1}gt)\right)\overline{\beta(g)}$

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

 $\frac{1}{\left\lvert G\right\rvert\left\lvert H\right\rvert}\sum_{% \begin{subarray}{c}t\in G\\ g\in G\\ t^{-1}gt\in H\end{subarray}}\alpha(t^{-1}gt)\overline{\beta(t^{-1}gt)}=\frac{1% }{\left\lvert G\right\rvert\left\lvert H\right\rvert}\sum_{h\in H}\sum_{% \begin{subarray}{c}t\in G\\ g\in G\\ t^{-1}gt=h\end{subarray}}\alpha(h)\overline{\beta(h)}$

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 $\left\lvert G\right\rvert$ times by the sum. Thus the sum is equal to

 $\frac{\left\lvert G\right\rvert}{\left\lvert G\right\rvert\left\lvert H\right% \rvert}\sum_{h\in H}\alpha(h)\overline{\beta(h)}=\frac{1}{\left\lvert H\right% \rvert}\sum_{h\in H}\alpha(h)\overline{\beta(h)}=\langle\alpha,\beta% \negmedspace\downarrow_{H}^{G}\rangle_{H}$

Title proof of Frobenius reciprocity ProofOfFrobeniusReciprocity 2013-03-22 18:36:23 2013-03-22 18:36:23 rm50 (10146) rm50 (10146) 5 rm50 (10146) Proof msc 20C99