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For every prime $p$ let ${\bf Z}_{(p)}$ be the subgroup of $({\bf Q}, 0, +, -)$ consisting of fractions $a/b$ whose denominator is not divisible by $p$.

Let $Z = \prod_p {\bf Z}_{(p)}$ where the product is finite support.

Is there an elementary embedding ${\bf Z} \to Z$?

I think the answer is no.

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    By the way, a finite support "product" is usually instead called a _direct sum_. – 2017-02-05
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    There's an unfortunate pattern of reasonable model theory questions about algebraic structures being downvoted and closed when they're tagged with "non-logic" tags like (abstract-algebra) and (group-theory), likely because most people who follow those tags don't know what terms like "elementary embedding" mean. If I were you, I would either not use those tags or point out explicitly when you use terms from logic. – 2017-02-05
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    Also, it wouldn't hurt to add more to the question. Just putting "I think the answer is no" doesn't show much effort. If you think the answer is no, you must have *some* reason, and writing down that reason can be a good start toward a solution. This apparently lack of effort could be another reason for the downvotes. – 2017-02-05

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Hint: Let $f:\mathbb{Z}\to Z$ be any homomorphism. Show that there must be some $n>1$ such that $f(1)$ is divisible by $n$ in $Z$, and use this to show $f$ is not an elementary embedding.