Find all the homomorphisms from S4 to Z2?

Find all the homomorphisms from S4 to Z3?

 

Where do I start on this one? I know that one is that everything can be mapped to the identity in Z2 or Z3, but there where do I go?

 

Help!

Pick generators of Sn, hint transpositions generate, all transpositions are conjugate.

We haven't done conjugate classes yet, so I don't think I can use that information to prove it.

 

This is what I got so far...

In the first part, I found two homomorphisms. All the elements of S4 mapped to the identity. And another one being, all the even permutations to 0 and odd permutations to 1. THe kernels being S4 and A4.

 

The second part I found one permutation, all elements in S4 going to the identity.

 

How do I prove that these are the only ones? My prof hinted that know the composition of Z2, that this are two subgroups would be a good start, but that doesn't make a ton of sense to me.

 

Any suggestions? Please help!!!

You do know that transpositions generate Sn? They must get sent to elements of order 1 or 2. so....

So let's add some more detail:

 

S4 to Z_2 (written additively to match your convention).

 

there's the trivial homomorphisms everything gets sent to 0

 

suppose now that (12) gets sent to 1. Since (123) must get sent to 0 ((123) has order 3, so its image has order divisible by 3) but (123) = (12)(23) so it must be that (23) gets sent to 1 as well. By similar logic (14) must be sent to 1, and similalry we see all 2-cycles must be sent to 1. This describes the homomorphism: s in S_4 is sent to 1 if it is an odd permutation 0 otherwise.