Abelian groups \(\aleph_ 0\)-categorical over a subgroup (Q2641294)

Consider P(x) a unary predicate and let \(L(P)=\{0,-,+,P(x)\}\). Let a pair (A,B) be a pair of Abelian groups such that \(B\subseteq A\). A pair (A,B) is an L(P)-structure where the operations and the constant 0 are as usual and P(x) stands for \(x\in B\). We say that an Abelian group A is \(\aleph_ 0\)-categorical over its subgroup B if for any countable pairs (C,D), \((C',D)\) which are L(P)-elementarily equivalent to (A,B) there is an isomorphism \(\phi: C\to C'\) such that the restriction of \(\phi\) to D is the identity. Let \(mA=\{ma\); \(a\in A\}\) for A an Abelian group and \(m\in {\mathbb{N}}\). Using the extension of isomorphism technique of \textit{I. Kaplansky} and \textit{G. W. Mackey} [Summa Brasil. Math. 2, 195-202 (1951; Zbl 0054.018)], it is shown that A is \(\aleph_ 0\)-categorical over B if and only if A/B is bounded and \(B\cap p^ nA\) is L-definable in B (without parameters), for all primes p and natural number n. Extending this result we give an axiomatization of such theories in terms of some first-order invariants. As a corollary we get a new proof of Hodges' Decomposition Theorem [\textit{W. Hodges}, Abelian groups, preprint (April 1986), Corollary 2.2], namely if A is \(\aleph_ 0\)-categorical over B then (A,B) can be decomposed as \((A_ 1,B)\oplus (B,0)\), where \(A_ 1\) is \(\aleph_ 0\)-categorical over B and it is minimal (up to isomorphism) with this property. We show in the last section that for A torsion-free, A is \(\aleph_ 0\)-categorical over B if and only if \(B=mA\) for some \(m\in {\mathbb{N}}\).