Surjective homomorphism from $\mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z} \times (\mathbb{Z}/2\mathbb{Z})$

Question

I have been trying to derive a surjective homomorphism from $\mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z} \times (\mathbb{Z}/2\mathbb{Z})$.

What are some examples of something like this? I require an additional condition that $\phi(x,y) = (0,z)$ (where $z$ is 0 or 1) if and only if $y = -2x$.

So far I have $\varphi(x,y) = (2x+y, y\text{ mod 2})$ which meets my condition but is not surjective, any way I can fix this?

Answer 1

Along the lines of the map you tried, first consider $g = \pmatrix{2 & 1 \\ 1 & 1}\in SL_2(\mathbb{Z})$. Since $\det g =1$, it's a surjection $\mathbb{Z}^2 \to \mathbb{Z}^2$, and clearly $g(x, y) = (0, z)$ for some $z\in \mathbb{Z}$ iff $y = -2x$.

Answer 2

In my opinion, the simplest approach to this is:

• Find all homomorphisms $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}/2 \mathbb{Z}$
• Check which of them satisfy the conditions you need.