Quotient group(Factor group)

Question

Prove that the quotient group $\frac{Z\times Z\times Z}{<(1,1,1)>}$ is an infinite, non-cyclic group.

Here Z is the group of integers with operation of addition, $<(1,1,1)$> is the subgroup of $Z\times Z\times Z$ generated by the element $(1,1,1)$

EDIT (1)-

I've been able to prove the infiniteness of the group. Here is how I have done it : Observe that < [(1,0,0) + < (1,1,1) >]> is an infinite cyclic subgroup of the given group in the question. hence the group in the question is also infinite.

However I am stuck on proving the non-cyclic part. I'm getting that < [(1,0,0) + < (1,1,1) >] > and < [(0,1,0) + < (1,1,1) > ] > are two cyclic subgroups, whose intersection is {e} that is identity. Can it lead me to anywhere?

Answer 1

I would reason like this. A set of representatives for your quotient is $$\{(0,n,m)+\langle(1,1,1)\rangle\ |\ n,m\in\mathbb{Z}\}$$ since $(a,b,c)+\langle(1,1,1)\rangle=(a-a,b-a,c-a)+\langle(1,1,1)\rangle$ so $$\frac{\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}}{\langle(1,1,1)\rangle}\cong\mathbb{Z}\times\mathbb{Z}$$

Can you show $\mathbb{Z}\times\mathbb{Z}$ is not cyclic?

(Assume it's generated by $(n,m)$ and find a contradiction.)