I am very sorry that I deleted my previous post on this same topic and specially to **@lhf**, since I forced him/her to delete his/her answer.

> 
> Prove that the product of two infinite cyclic groups is not infinite cyclic.
> 
> 
> 

The part I did:

$G=<a>$ and $G'=<b>$. The product group is $G\times G'$, where mulplication rule is $$(a^{k\_1},b^{l\_1}),(a^{k\_2},b^{l\_2})\rightsquigarrow (a^{k\_1+k\_2},b^{l\_1+l\_2})$$ where $k\_1,k\_2,l\_1,l\_2\in \mathbb{Z}$

Now suppose the cyclic group $G\times G'$ is generated by $(a^k,b^l)$

Then for any $m,n\neq 0$, $(a^m,b^n)=(a^{ku},b^{lu})\Rightarrow \frac{k}{l}=\frac{m}{n}$, which can not be true since $k,l$ are fixed and $G,G'$ are infinite cyclic group.

But then I am **stuck**. What should be the lat argument to end this proof!