I need to find the the possible homomorphisms from $A_n$ (Alternating group) for $n \geq 5$ to $C^*$ (the group of non-zero complex numbers).
Fact: $A_n$ for $n \geq 5$ is simple
For any group homomorphism from a simple group to an arbitrary group, the kernel being normal is either trivial or the group itself.
Also we see that even cycles can not map to -1 but only to 1.
So the kernel is the group $A_n$ for $n \geq 5$ itself which in turn implies the homomorphism to be trivial by the first isomorphism theorem.
Is my argument valid?
Any suggestions, improvements or an easier approach would be of much help.