Group homomorphism from $S_3$ to $A_3$

Question

How many number of group homomorphisms are there from $S_3$ to $A_3$ ??

In case of $Z_m$ to $Z_n$, I can map 1 to an element whose order divides order of both $Z_m$ and $Z_n$ (m and n).

What method should be applied to non abelian groups??

$S_3$ is of order 6 , $A_3$ is of order 3 Identity should be mapped to identity. And how can we conclude about mapping of other elements in $S_3$ to $A_3$..

Answer 1

$S_3$ is generated by ??

It does not help to consider generating set as full $S_3$. What is the smallest generating set can you think of for $S_3$?

Any morphism is determined by generators of its domain. Once you know generators you will know possible images in $A_3$ depending on order.

Answer 2

Note that if $\phi : G \to H$ is a homomorphism, then $|\phi (a)|$ divides $|a|$ for all $a \in G$.

Now, let $\phi : S_3 \to A_3$ be a homomorphism.

What does this imply on the order of the image of $2$-cycles under $\phi$? (What are the possible orders in $A_3$?)

Then, recall that the $2$-cycles generate $S_3$, and conclude that the only homomorphism $S_3 \to A_3$ is trivial.