Prove that the product of two infinite cyclic groups is not infinite cyclic.
Let $G=<a>$ and $G'=<b>$. The product group is $\{a^mb^n: m,n\in\mathbb{Z}\}=<a^kb^l>$
Now, for given $m,n\in\mathbb{N}$, we can find $u$ such that $(a^kb^l)^u=a^mb^n$. Then $u=\frac{m}{k}=\frac{n}{l}\Rightarrow \frac{k}{l}=\frac{m}{n}$, which is true for any $m,n\in\mathbb{N}$, which can not be true, since $k,l$ are fixed, so the ratio is. And then I am stuck. What to do next???
Another question from my curiosity: We can take $G=<e^{i\pi \sqrt{2}}>$ and $G'=<e^{i\pi \sqrt{3}}>$. What is the product group? It should be $<e^{i\pi \theta}>$, where $\theta$ is rational. But what is $\theta$?