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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 $V$.

Romeo
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    What have you tried? Can you identify a subgroup of $A_4$ that is isomorphic to $V$? – rogerl Jan 15 '20 at 17:34
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    You know that in general if $H$ is a normal subgroup of $G$ there is a homomorphism of $G$ onto $G/H$ with kernel $H$. The proof of that fact constructs an explicit homomorphism. – David C. Ullrich Jan 15 '20 at 17:34
  • Use the answers here and restrict the homomorphism $f:S_4\to S_3$ to the subgroup $A_4$, when the image is automatically $A_3\simeq \Bbb{Z}_3$. – Jyrki Lahtonen Jan 16 '20 at 06:13

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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 $\Bbb Z_3 = \{0,1,2\}$.

Ben Grossmann
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    I suspect that there's a nicer interpretation here if we think of the corners as being corners of a tetrahedron rather than of a square. – Ben Grossmann Jan 15 '20 at 18:01
  • Thanks for the idea, but I have tried this defining the function practically specifying how each element of each coset in $A_4/V$ is sent to $\mathbb{Z}_3$... I consider that this is not a "natural way" of doing this... there is another way? – Romeo Jan 15 '20 at 20:18
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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.