$f: G \to \mathbb{C}^*$ is a homomorphism. Show that the sum $\sum f (g) = 0$ or $n$

It is not necessary that $G$ be abelian, to wit:

If

$f(g) = 1, \; \forall g \in G, \tag 1$

then clearly

$\displaystyle \sum_{g \in G} f(g) = n, \tag 2$

since

$o(G) = n; \tag 3$

if

$\exists h \in G, \; f(h) \ne 1, \tag 4$

then since

$hG = G, \tag 5$

we have

$$\begin{align} \sum_{g \in G} f(g) &= \sum_{g \in G} f(hg) \\ &= \sum_{g \in G} f(h)f(g) \\ &= f(h)\sum_{g \in G} f(g); \tag 6 \end{align}$$

with $f(h) \ne 1$ this forces

$\displaystyle \sum_{g \in G} f(g) = 0. \tag 7$

$OE\Delta$.