Homomorphism from the alternating group $A_4$ to $\mathbb{Z}_3$

Question

Well, my question is how to construct an explicit homomorphism from the alternating group $A_4$ onto $\mathbb{Z}_3$, since it is known that the quotient group $A_4/V\cong \mathbb{Z}_3$, $V$ being the Klein group, I would like to find such a homomorphism with kernel $V$.

Answer 1

Of course, it's easy to build such a homomorphism by deciding what happens to each coset of $V$, the subgroup of double-transpositions (i.e. the only Klein group in $A_4$). That being said, such a definition isn't particularly intuitive.

There are two such homomorphisms, I'll describe one of them geometrically. Identify $A_4$ with the permutations of the corners of a square; label the corners $1,2,3,4$ clockwise. Note that each element of $A_4$ is either a double-transposition, or fixes a corner and rotates the remaining elements. For any $g \in A_4$, we define $$ \phi(g) = \begin{cases} 0 & g \text{ is a double-transposition}\\ \\ 1 & g \text{ fixes an odd number and rotates}\\ & \text{the remaining elements clockwise or }\\ & \text{fixes an even number and rotates the }\\ & \text{remaining elements ccw}\\ \\ 2 & g \text{ fixes an even number and rotates}\\ & \text{the remaining elements clockwise or }\\ & \text{fixes an odd number and rotates the }\\ & \text{remaining elements ccw}\\ \end{cases} $$ where $\Bbb Z_3 = \{0,1,2\}$.

Answer 2

Hint:

Your Klein four group is $\{e, (12)(34), (13)(24), (14)(23)\}$. Now use the canonical projection.

Note: there is another one, since there is a nontrivial automorphism of $\Bbb Z_3$, gotten by composition.