I want to know if the group $G=\mathbb Z \times \mathbb Z$ can be written as union of finitely many proper subgroups of it ?
It is clear that $\mathbb Z$ can't be written as union of finitely many proper subgroups as the subgroups are of the form $n \mathbb Z$ for some integer $n$ and there are infinitely many primes in $\mathbb Z.$
My way to think: If possible $G= H_1 \cup \cdots\cup H_r $ where $r>1$ and $H_i's$ are proper subgroups of $G.$ Now considering the projection maps $\pi_1$ and $\pi _2$ on $G$ there exist $i$ and $j$ such that $\mathbb Z=\pi_1(H_i)$ and $\mathbb Z=\pi_2(H_j).$ I can't complete my arguments after that. Any helps will be appreciated. Thanks.