Prove that $\mathbb{Z}_2 \times \mathbb{Z}_2$ is not cyclic

Question

My attempt: $\mathbb{Z}_2 $has elements of the form $\{1,x\}$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ has elements of the form $\{(1,1),(1,x),(x, 1),(x, x) \}$ order of $(1,1)=1$, order of $(1,x) ,(x, x)$ and $(x, 1)$ is $2$. Since none of the elements have order =4 . It is not a cyclic group as there is no generator. Am I correct?

Answer 1

Looks good to me. There can be no single generator because no element has high enough order, thus the group is not cyclic.