Homomorphism from group of integers modulo $4$ to the Klein four group

Question

Let $G=\mathbb{Z}_4$, the group of integers modulo $4$, and let $H$ be the Klein four group, let $f: G \rightarrow H$ be a homomorphism. Why does the kernel of $f$ contain the element $2$ of $G$?

Answer 1

Using the homomorphism property, $$f(2)=f(1+1)=f(1)^2.$$ What is $f(1)^2$ in the Klein 4 group? (Try out all the different possible values for $f(1)$ if you're not sure.)

Answer 2

Let's find the possible homomorphisms.

First we require $f(0) = 1_H$

So now we need $1_H = f(0) = f(1+3) = f(1)f(3)$. So $f(1) = f(3)$.

Now $1_H = f(3)^2 = f(3 + 3) = f(2) = f(1+1) = f(1)^2 = 1_H$.

So all homomorphisms are of the form $\{0, 1, 2, 3\} \mapsto \{1_H, h, 1_H, h\}$ where $h$ is any element in the Klein 4 group.