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 $A_3$..
Best Answer
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.