Is every Abelian group isomorphic to the external direct product of two cyclic groups?

Question

Is every Abelian group isomorphic to the external direct product of two cyclic groups?

I know that this statement holds true for finite cyclic or non cyclic groups, like $U(n)$ for example, but I am not sure if it's true for infinite abelian groups. So I tired to find a counter example, by assuming there is an isomorphism between $\Bbb R$ and the external product of $\Bbb Z$ and $\Bbb Z$! But I don't know if $\Bbb Z \times \Bbb Z$ is cyclic.

And also the order of both of them isnt the same, though both are infinite! I am confused on these two things , maybe an example and explanation would make things clear ?

Answer 1

Cyclic groups are at most countable, so the direct product of two cyclic groups is at most countable. So $\mathbb{R}$, which is uncountable, can't be the direct product of two cyclic groups.

But actually the statement isn't true for finite groups either. Consider the direct product of three copies of $\mathbb{Z}_2$. This is not the direct product of any two cyclic groups.

Answer 2

Your initial assumption is incorrect. There are lots and lots of finite abelian groups which aren't products of two cyclic groups.

Secondly, $\Bbb Z\times \Bbb Z$ is not cyclic. If it were, then any homomorphic image would be cyclic. But $\Bbb Z_2\times\Bbb Z_2$ is not cyclic.

(Interestingly enough, there is only one infinite cyclic group, up to isomorphism. That's $\Bbb Z$.)