$\mathbb{Z}_{22}$ surjects onto $\mathbb{Z}_2$

Question

Show that $\mathbb{Z}_{22}$ surjects onto $\mathbb{Z}_2$.

I assume that we should use isomorphism theorems here or maybe quotients maps?

I tried finding homomorphism for the first isomorphism theorem but I couldn't find any...

And ideas?

Answer 1

Hint: Prove directly that $x \bmod 22 \mapsto x \bmod 2$ is a well-defined surjective homomorphism.

More generally, if $d$ divides $n$, then $x \bmod n \mapsto x \bmod d$ is a well-defined surjective homomorphism $\mathbb{Z}_{n} \to \mathbb{Z}_{d}$.

Answer 2

You want exactly half of the elements of the domain in the kernel of your map. Think about the parity of the elements.

Answer 3

Others have already pointed out how you can create a surjective homomorphism. But $$ \textbf{if} $$ you are just looking for a function from $\mathbb{Z}_{22} \to \mathbb{Z}_2$, you could create any function you wanted. For example $$ f(x) =\begin{cases} 0 & \text{if } x = 0 \\ 1& \text{if } x\neq 0 \end{cases} $$ will be a surjective function.