Finding homomorphism for a specific kernel

Question

I'm trying to find an homomorphism from $Z_{4} \times Z_{6}$ to $Z_{4} \times Z_{2}$ with kernel $\langle (0,2) \rangle$.

How can I show there exists such homomorphism and how do I actually find the homomorphism?

I know $Z_{4} \times Z_{6}$ is abelian and $\langle (0,2) \rangle $ is a normal subgroup of $Z_{4} \times Z_{6}$, but I'm pretty much stuck there...

Answer 1

Hint - look at the map $\phi(a,b) = (a, b \pmod 2)$.

Answer 2

It's the canonical projection. When $N\trianglelefteq G$, define $p:G\to G/N$ by $p(g)=gN$.

Since $\langle(0,2)\rangle\cong \Bbb Z_1×\Bbb Z_3$, we have $(\Bbb Z_4×\Bbb Z_6)/(\Bbb Z_1×\Bbb Z_3)\cong\Bbb Z_4/\Bbb Z_1×\Bbb Z_6/\Bbb Z_3\cong\Bbb Z_4×\Bbb Z_2$.