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Is there any fairly easy way of showing a group is elementarily equivalent to the additive group of the integers?

I've found a simple characterization here: A ‘natural’ theory without a prime model, but the proof in Szmielew's paper is quite long and much more general, while I'm looking for something more elementary.

Specifically, I'd like to show that the subgroup of rationals generated by fractions of the form 1/p for p prime is equivalent to integers, but a more general, relatively simple solution would be appreciated.

edit: As pointed out in the comments, i mean the additive group of rationals (clearly, since for the multiplicative group the fractions would generate the entire group, and it's certainly not equivalent to integers, whether it's multiplicative or additive), and the subgroup can also be characterized as the group of fractions with squarefree denominators, while elementary equivalence is a concept from model theory (as indicated in tags).

szmielew's paper considering equivalence classes of abelian groups can be found here: matwbn.icm.edu.pl/ksiazki/fm/fm41/fm41122.pdf , but it's from the 50's, making it quite hard to read due to outdated language and very apparent lack of modern latex.

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    Do you mean subgroup of the rational numbers under multiply? If so, you can apply unique factorization theorem of integers to your question.2012-04-22
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    @jerrysciencemath: We can generate additive groups which are much larger than the rationals which are elementarily equivalent to the integers (e.g. ultrapowers of the integers).2012-04-22
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    @jerrysciencemath: No I think he means the additive group of rational numbers of the form $n/m$ where $m$ is square-free.2012-04-22
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    What do you mean by 'elementarily equivalent'? Does it mean isomorphic?2012-04-22
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    @jerrysciencemath No, this is a concept from model theory. See http://en.wikipedia.org/wiki/Elementary_equivalence.2012-04-22
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    @groovin: could you provide us with the electronic copy of the article since I don't have a subscription?2012-04-22
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    i meant the additive group and elementarily equivalent, just the way henning makholm and jim belk have indicated. the article i linked does not contain proof, altough the article (which is quite dated) containing the proof in question is available without subscription, free of charge, right here: http://matwbn.icm.edu.pl/ksiazki/fm/fm41/fm41122.pdf .2012-04-22

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