Show that the infinite cyclic group is not isomorphic to a direct product of two nontrivial cyclic groups.

Show that the infinite cyclic group (for example, $\mathbb{(Z,+)}$) is not isomorphic to a direct product of two nontrivial cyclic groups.

I'm really having trouble with this question. How would you show that such an isomorphism doesn't exist? Would appreciate if someone could answer in detail as it would really deepen my understanding of cyclic groups.

• As a general rule, I find that "answering in detail" does not deepen your understanding. Working it out for yourself is best - you only learn when you are getting stuck, and you learn best when you make slow, slow progress. Simply getting an answer skips the whole "slow progress" thing and can be detrimental in the long term. – user1729 Apr 19 '18 at 18:48

The infinite cyclic group $\Bbb{Z}$ cannot be isomorphic to $\Bbb{Z}\times \Bbb{Z}$, because the latter cannot be cyclic. It also cannot be isomorphic to $\Bbb{Z}\times \Bbb{Z}/n\Bbb{Z}$ for the same reason; and finally not to $\Bbb{Z}/m \Bbb{Z}\times \Bbb{Z}/n\Bbb{Z}$, because this is finite. But these are the only possibilities of non-trivial cyclic factors in a direct product.