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here: matwbn.icm.edu.pl/ksiazki/fm/fm41/fm41122.pdf , but it's from the 50's, making it quite hard to read due to outdated language and very apparent lack of modern latex.