The average value of irreducible character of a non-trivial finite group

Question

Let $G$ be a non-trivial finite group. Let $\chi$ be an irreducible character of the group $G$. Find $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}$$

I try. But I think that I am wrong. $$G=C_{i_1}\oplus C_{i_2}\oplus\ldots C_{i_k},$$ as $C_{i}$ is a cyclic group If $\chi$ is 1-dimensional character then $\sum_{g \in C_{m}} {\chi( g)}=0.$ Thus $$\frac{1}{|G|}\sum_{g \in G} {\chi( g)}= \frac{k}{|G|}.$$ Am I on the right path?

There are not answer or hints in our book

Answer 1

Here is an extension of the comment by mt_:

Consider the definition of the inner product on the characters $\langle \chi,\psi\rangle = \frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\psi(g)}$.

Can you get the sum you are looking as an inner product between suitable characters?

Answer 2

Apply the orthogonality relations.

Alternatively, consider an irreducible representation $\mathscr{X}$ affording $\chi$. Let $A = \sum_{g \in G} \mathscr{X}(g)$. Prove that either $A = 0$ or $\mathscr{X}$ is the trivial representation.