Originally Posted by
Taluivren Hi, what exactly is not clear?
by definition, a cyclic group is a group that can be generated by a single element. As I understand, you ask whether $\displaystyle \mathbb{Z}\times\mathbb{Z}$ is cyclic or not?
If $\displaystyle \mathbb{Z}\times\mathbb{Z}$ were cyclic, then there would be an element $\displaystyle (a,b) \in \mathbb{Z}\times\mathbb{Z}$ such that every element $\displaystyle (c,d) \in \mathbb{Z}\times\mathbb{Z}$ can be written as $\displaystyle (c,d) = n(a,b) = (na,nb)$ for some integer $\displaystyle n$. The proof in the previous post shows that if you choose any $\displaystyle (a,b) \in \mathbb{Z}\times\mathbb{Z}$ then there exists an element from $\displaystyle \mathbb{Z}\times\mathbb{Z}$ such that no such $\displaystyle n$ exists. This contradicts the statement that $\displaystyle \mathbb{Z}\times\mathbb{Z}$ is cyclic.