proof of Frobenius reciprocity
We prove the slightly more general result
Here we use to refer to the induction (http://planetmath.org/InducedRepresentation) to of a class function on , and to refer to the restriction (http://planetmath.org/RestrictionRepresentation) of a class function on to one on .
Since is a class function, this is the same as
Clearly for every there is a unique with , so every element of is counted times by the sum. Thus the sum is equal to