TITLE: One-dimensional representations of S5
QUESTION [5 upvotes]: 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 $S_n$

REPLY [5 votes]: 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.