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Stable actions of periodic abelian groups on abelian groups. (English) Zbl 0745.20048

Let \(A\) and \(B\) be abelian groups, \(f\) be an action of \(B\) on \(A\). The centralizer \(C^ B(A_ 0)\) of \(A_ 0\subset A\) is the subgroup \(\{b\in B\mid f_ b(a)=a\) for every \(a\in A_ 0\}\) of \(B\). The center \(Z_ 1^{B_ 0}(A)\) with respect to \(B_ 0\subset B\) is \(\{a\in A\mid f_ b(a)=a\) for all \(b\in B_ 0\}\). \(Z^{B_ 0}_{i+1}(A)\) is \(\{a\in A\mid f_ b(a)-a\in Z_ i^{B_ 0}(A)\) for all \(b\in B_ 0\}\). The main result is the following: Let \(B\) be a \(p\)-group, \(f\) be an action (of \(B\) on \(A\)) with the minimal condition on centralizers and there is an upper bound \(n\) for the lengths of chains of centralizers. Then there is some subgroup \(B_ 0\) of \(B\) such that \(B/B_ 0\) has finite Prüfer-rank and \(Z_ 1^{B_ 0}(A)\supset Ap^ n\). Corollary: Assume that in some stable structure abelian groups \(A\) and \(B\) and an action of \(B\) on \(A\) are defined, where \(B\) is periodic. Then there is some subgroup \(B_ 0\) of \(B\) such that \(B/B_ 0\) has finite Prüfer- rank and the action of \(B_ 0\) on \(A\) is nilpotent.

MSC:

20K10 Torsion groups, primary groups and generalized primary groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
03C45 Classification theory, stability, and related concepts in model theory
20A15 Applications of logic to group theory
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