One-dimensional representations of S5

Question

The only one-dimensional representations of $S_5$ are the trivial representation and the sign representation. Why are these the only ones?

Here's what I've got so far: the image of any one-dimensional representation is abelian. If $\rho:G \rightarrow H$ is a homomorphism and $H$ is abelian, then $H$ is either the trivial group or $C_2$ (why?).

Source: Artin's Algebra 7.1.4. Is the fact that this is $S_5$ just a red herring? My initial instinct was to use simplicity of $A_5$, but it seems like these are the only one-dimensional representations for any $S_n$

Answer 1

The derived subgroup is $A_5$, which has index 2. A homomorphism $G\to H$ has abelian image if and only if $G'$ is in the kernel. A linear character is a group homomorphism to the abelian group of units of the field.

More generally, $A_n$ is the derived subgroup of $S_n$, and for nice enough fields the group of linear characters of $G$ is isomorphic to $G/G'$. The complex numbers are always nice enough.

Answer 2

For the sake of completeness for future readers I will add an elementary solution (which is also valid for $n<5$).

Observe that since any permutation is a product of transpositions, it suffices to show that for any representation $\pi \colon S_n \to \mathbb{C}^\times$ either $\pi$ takes the value $1$ on all transpositions or $-1$ on all transposition. Indeed, since every transposition has order $2$, it is certainly the case that $1$ og $-1$ are the only options. On the other hand, all transpositions are conjugate and so $\pi$ must take the same value on all of them.