Find all homomorphism from $S_4$ to $\mathbb{Z}_2$ I am supposed to find all group homomorphisms from $S_4$ to $\mathbb{Z}_2$, I tried to check if $S_4$ was cyclic then it would be easy but of course it isn't.
I am not looking for a solution since this is a homework and I want to solve it myself, but I would appreciate some hints.
 A: Some hints that may or may not help:


*

*The kernel of any homomorphism $\phi:S_4 \rightarrow G$ for any $G$ will be a normal subgroup of $S_4$, and there are only two such subgroups.

*$\phi(e) = e$ for all group homomorphisms.

*In this case, any non-trivial homomorphism is surjective.

*The isomorphism theorem: $Im(\phi) \cong S_4/ker(\phi)$.

A: Here are some hints: suppose $\phi:S_4\to\Bbb Z_2$ is a homomorphism of groups.


*

*Let $\tau$ be a transposition, show that if $\phi(\tau)=0$ then $\phi$ is trivial.



 The transpositions generate $S_4$, and all transpositions are conjugate to each other.

Now suppose $\phi$ is non-trivial. By what precedes, $\phi(\tau)=1$ for all transpositions.


*

*Show that $\phi$ is the signature morphism.



 If one identifies $(\Bbb Z_2,+)$ with $(\lbrace -1,+1\rbrace,\times)$, which is the usual target of the signature morphism $\epsilon$, we have that $\epsilon(\sigma)=k\mod 2$ (where $\sigma$ is the product of $k$ transpositions). Indeed, the usual formula for the signature is $\epsilon(\sigma)=(-1)^{k}$, $k$ as above. From the first bullet point we know that $\phi(\tau)=1$ for all transpositions $\tau$, so that if $\sigma$ is a product of $k$ transpositions, then $\phi(\sigma)=k\mod 2$, so that $\phi$ is the signature morphism.

