The elementary theory of abelian groups

Ill this paper we make a compretiensive survey of the first-order properties o f abelian groups. The principal method is the investigation of saturated abelian groups. As a result of our determination of the structure o f saturated groups we are able to give new model-theoretic proofs of the results o f W, Szmietew [ 12 ] ; moreover we obtain new results on the existence o f saturated models of complete theories o f abelian groups; and we also generalize our results to modules over Dedekind domains. One of tile principal results of Szmielew is the determination o f grouptheoretic invariants which characterize abelian groups up to elementary equivalence (The decidability o f the theory of abelian groups follows relatively easily from this result), Now elementarily equivalent saturated groups o f the same cardinality are isomorphic; so our method is to look fi_~r invariants which characterize saturated abelian groups up to isomorphism. We prove that any ~:-saturated group A(tc >_ ~ ) is built up in a specified way from the groups ZQ~'), Zp, Z(p n) and ~ and that the number o f copies of these groups which occur are determined by the elementarily definable dimensions dim(p n! A[p] ), dim(p nl A / o n A ), and dim(p n=l A[p ] /pnA [p] ) and by the exponent o f A (for explanations o f the notation and more details, see § l ). These dimensions, which arise