I am currently teaching myself some representation theory and I just ran into the following statement:
The number of linear representations of $\mathcal{S}_n$ is equal to the order of the Abelianization of $\mathcal{S}_n$, i.e. $\left\vert \mathcal{S}_n / \mathcal{S}_n'\right\vert$, where $\mathcal{S}_n'$ is the derived subgroup of $\mathcal{S}_n$ which is $A_n$.
I want to why it is true that the number of distinct linear representations is equal to the order of the Abelianization and I can not seem to find a proof anywhere!
There's nothing special about $G=S_n$ here; any group will do. Since $\mathbb C^\times$ is commutative, a group homomorphism $G \to \mathbb C^\times$ has $G'$ in the kernel, so linear representations of $G$ are in one-to-one correspondence with linear representations of $G/G'$.