Fourier analysis on finite abelian groups Can someone help me show if $f$  is a character of a finite abelian group then for all $a\in G$,
$$\sum_{[f]}f(a)\stackrel{}{=}
\begin{cases}
|G|     & \text{if $ a$ is the identity} \\
0 & \text{otherwise}
\end{cases}$$
Where the sum runs over all characters of $G$,
I was thinking of using the fact that the characters themselves form an abelian group under multiplication, so that I have$$f_k(a)\sum_{[f]}f(a)=\sum_{[f]}f(a)f_k(a)=\sum_{[f]}f(a)$$
So that, $$(1-f_k(a))\sum_{[f]}f(a)=0$$
But I would need $\exists k|f_k(a)\ne 1, \forall a \in G, a\ne e$
In order to divided both sides by $(1-f_k(a))$, which I can't seem to get either.
 A: The proof (which is very standard) can for instance be found in $\S 4.3$ of Appendix B of these notes.
If you take a look, you'll see that you're on the right track, but perhaps you've been stymied by a quantifier error.  You want to show your identity for each fixed nonidentity element $a \in G$.  So you don't need to find a character which is nontrivial on all nonidentity elements (which is not always possible; consider e.g. $G = Z/2 \times Z/2)$.  You just need that each nonidentity element is not killed by at least one character on $G$.  There is still something to show here, and the notes explain this point carefully.  (It follows from something called the Character Extension Lemma...)
A: Your proof strategy is correct. To prove that there is a character with $\chi(a)\neq 1$ look at Serre's A course in arithmetic:
Proposition 1.1.: Let $H$ be a subgroup of $G$ (finite), then every character $\chi$ on $H$ extends to a character on $G$.
Proof (Sketch): By induction on $(G:H)$. Let $H$ be a non-trivial subgroup, $x\notin H$, and $n$ minimal such that $x^n\in H$. If $\chi(x^n)=t$ choose any $w\in\mathbb{C}^{*}$ with $w^n=t$ and define $\chi'(x)=w$. This gives an extension of $\chi$ to the subgroup of $G$ generated by $H$ and $x$.
A: This is the so called column orthogonality relation, applied to the conjugacy classes of $a$ and $1$. 
The proof can be found in any book on representation theory. One first shows that the irreducible representations are a basis for the vector space of class functions, then one expresses one of the standard basis class functions (one that is 1 on one class and 0 on all other classes) as a sum of irreducible characters. One can compute the coefficients explicitly in this sum and then fill in $a$ in the relation that you obtain (noting that $|g^G|=1$, since $G$ is abelian).
