proof of Frobenius reciprocity

We prove the slightly more general result

Theorem 0.1.

If G is a finite groupMathworldPlanetmath with subgroupMathworldPlanetmathPlanetmath H, α a class function on H and β a class function on G, then


Here we use HG to refer to the inductionMathworldPlanetmath ( to G of a class function on H, and HG to refer to the restrictionPlanetmathPlanetmathPlanetmath ( of a class function on G to one on H.


Since β is a class function, this is the same as


Clearly for every hH,tG there is a unique gG with t-1gt=h, so every element of H is counted |G| times by the sum. Thus the sum is equal to


Title proof of Frobenius reciprocity
Canonical name ProofOfFrobeniusReciprocity
Date of creation 2013-03-22 18:36:23
Last modified on 2013-03-22 18:36:23
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 5
Author rm50 (10146)
Entry type Proof
Classification msc 20C99