Construct a non trivial homomorphism from $\mathbb{Z}_2\times\mathbb{Z}_4$ to $\mathbb{Z}_8$

I don't know how to solve this problem since the group $$\mathbb{Z}_2\times\mathbb{Z}_4$$ is not cyclic. I just know that $$\mathbb{Z}_2 \times \mathbb{Z}_4= \{ (0,0), (0,1),(1,0),(1,1),(0,2),(1,2),(0,3),(1,3) \}$$ and that the orders of these elements are:

$$(0,0)$$: $$1$$
$$(0,1)$$: $$4$$
$$(1,0)$$: $$2$$
$$(1,1)$$: $$4$$
$$(0,2)$$: $$2$$
$$(1,2)$$: $$2$$
$$(0,3)$$: $$4$$
$$(1,3)$$: $$4$$

Can anyone help me?

A homomorphism out of $$\mathbb{Z}_2 \times \mathbb{Z}_4$$ is determined by where it sends $$(1,0)$$ and $$(0,1)$$. To make it non-trivial, you just need to send one of those to somewhere that isn't $$(0,0)$$. Note that you must send $$(1,0)$$ to an element of order $$1$$ or $$2$$, and $$(0,1)$$ to an element of order $$1$$, $$2$$, or $$4$$, or the resulting function will not be a homomorphism.
• Thank you for your help.So,I just have to send (1,0) to 0 or 4 and (0,1) to 0 or 4 or 2 or 6 but without both being 0 at the same time.Right? If I choose for example f(1,0)=0 and f(0,1)=2 I want to show that f(0,0)=0$\in \mathbb{Z_8}$ in order f to be an homomorphism.So, f(0,0)=f(2,4)=2f(1,2).We know that (1,2)=(1,0)+2(0,1).How can I compute f(1,2)?Is it correct to say that f(1,2)=0+2$\cdot$2=4 ,so f(0,0)=2$\cdot$4=8=0 in $\mathbb{Z_8}$? I don't know if my thoughts are right. – Failousa Jan 4 at 19:43
• $f(1,2) = f(1,0) + 2f(0,1)$, since $f$ is a homomorphism, so $f(1,2) = 2(2) = 4$. You don't need to prove that $f(0,0) = 0$: you defined it to be that. – user3482749 Jan 4 at 19:46
Hint: If $$G_1$$, $$G_2$$ and $$G_3$$ are additive abelian groups, a homomorphism $$G_1\times G_2\to G_3$$ is of the form $$(x,y)\mapsto \alpha(x)+\beta(y)$$ where $$\alpha\colon G_1\to G_3$$ and $$\beta\colon G_2\to G_3$$ are (arbitrary) homomorphisms. This is not trivial so long as either $$\alpha$$ or $$\beta$$ is not trivial.
Can you find a nontrivial homomorphism $$\mathbb{Z}_2\to\mathbb{Z}_8$$?
• I didn't know that.Thanks for your help!Yes,I can find a nontrivial homomorphism $\mathbb{Z_2}\to\mathbb{Z_8}$ but can I have a proof of what you wrote? Or a link where I can find it? – Failousa Jan 4 at 19:58