I'm trying to understand a part of a proof of orthogonality relations for character groups and finite abelian groups, and I don't quite get this part from the below link:
http://www.ms.uky.edu/~pkoester/research/charactersums.pdf
In 2.3, Proposition 10, it is given that χ is not equal to χ_0. What exactly does it mean for a character to not be equal to another? Does this mean they are two different homomorphisms? For Proposition 10, is χ a fixed homomorphism in the character group of G?
How do they reach the conclusion that (χ(y) − 1), summation of (χ(y) − 1)χ(x) for every x in G = 0?
Thanks for your help.