How to distinguish two groups $(\mathbb{Z},+)$ , $(\mathbb{Z} \times \mathbb{Z},+)$ using first order logic? 
How can I distinguish two groups $(\mathbb{Z},+)$ and $(\mathbb{Z} \times \mathbb{Z},+)$ in first order logic?


The first one is cyclic and the second one is not but I can't find any thing in first order to prove they are not the same.

It might have been better to ask for a proof in the first-order theory of groups, instead of "in first-order logic", since after all ZF is a first-order theory. Not to be pedantic, but it seems this did lead to some confusion. Anyway:
Consider the sentence $\exists k \forall x \exists y (x=y+y\lor x=y+y+k)$.
Or, to say the same thing more colorfully: Note that "the subgroup of even elements has index $2$" is definable in the first-order theory of abelian groups.
