show G has a faithful representation of degree 1 over $\mathbb{C}$ iff it is cyclic

Question

Show that a finite group G has a faithful representation of degree 1 over $\mathbb{C}$ iff G is cyclic.

I saw the following related post, but I'm not sure if I fully understand it: Faithful representation implies group is cyclic.

If $G$ is cyclic, then suppose $a$ generates $G$. We can send $a^n$ to $n\in \mathbb{C}$ to define a faithful representation of degree 1.

If $G$ has a faithful representation of degree 1, we can define a G-invariant inner product over $\mathbb{C}$ as $\langle u,v \rangle = 1/|G|\sum_{g\in G} \langle gu, gv\rangle$ (I don't think dividing by $|G|$ is necessary). Why does this mean G can be regarded as a subgroup of $\mathbb{C}$? I know that every subgroup of the nonzero complex numbers is cyclic.

Answer 1

As Kenta points out, this is not a representation of $G$. Usually, a representation refers to a homomorphism $G\to \mathrm{GL}(V)$ for $V$ a vector space.

Here is a hint:

1. Suppose $G$ admits a faithful representation $\rho:G\to \mathrm{GL}(1,\Bbb{C})=\Bbb{C}^*$. As $\rho$ is faithful, we can identify $G$ with a finite subgroup of $\Bbb{C}^*$ using $\rho$.
2. Show that $G$ must be contained in the unit circle $S^1\subset \Bbb{C}^*$ (if there exists $g\in G$ with $\lvert g\rvert \ne 1$ what happens when we consider $\lim_{n\to\infty}\lvert g^n\rvert$?)
3. Classify finite subgroups of $S^1$.

The converse is easier; if $G$ is cyclic of order $n$, send a generator to $\exp(2\pi i /n)$.