Thread: Prove this mapping is a homomorphism

Here is the example:
The mapping from S_n to Z_2 that takes an even permutation to 0 and an odd permutation to 1 is a homomorphism.

-I have to prove that this example is a homomorphism.

I have looked at other homomorphisms and I understand how to prove them. They are pretty easy, such as determinants or derivatives. For some reason this one just eludes me, and I feel that there is a simple answer. i know that i have to prove phi(ab)=phi(a)phi(b) . this is an additive group so i think it would be phi(a+b)=phi(a)+phi(b) right? and i maybe have to do two cases one for even, one for odd? Thanks for help in advance.

no, the group operation in is composition, so what you want to prove is:

, for .

what this boils down to is proving:

even composed with even is even
even composed with odd is odd
odd composed with even is odd
odd composed with odd is even