Describe all the one-dimensional complex representations of a finite abelian group $G$. Deduce that the number of inequivalent degree $1$ complex representations of $G$ are equal to $|G|$.
attempt: By the fundamental theorem of abelian groups , $G$ can be expressed as a direct product of cyclic groups $G = C_1 \times ... \times C_n$,
where $|C_i| = n_i$. And recall every irreducible complex representation of $G$ is 1 dimensional (i.e a homomorphism between $G → \mathbb{C}^{\times}$.
Then for each generator $x_i \in C_i$ , then by the representation $\phi : G → \mathbb{C}^{\times}$, then $\phi(x_i)^{n_i} = 1$ ,so $n_i$ is a root of $1$. So we map each $x_i$ to a $n_i$ root of $1$ and extend this to all powers of $x_i$, the number of distinct homomorphism of $G$ into $\mathbb{C}^{\times}$ defined by this process equals $|G|$.
Can someone please help me? I need to know if this is fine, and if not please can someone please provide feedback or better approach. Thank you !
