Stable actions of torsion groups and stable soluble groups (Q1204450)

If \(G\) is a stable group (that is, a group whose elementary theory is stable) then, by the Baldwin-Saxl condition, there is a finite upper bound on the lengths of chains of centralizers \(C_ G(X_ 1) < C_ G(X_ 2) < \dots < C_ G(X_ r)\) where each \(X_ i\) is a subset of \(G\). The authors study such groups by means of the following key result. Let \(A\) be an abelian group (written additively) and \(B\) an abelian \(p\)- group (where \(p\) is a prime) which acts on \(A\) in such a way that there is an integer \(n\) such that every chain of centralizers \(C_ B(a_ 1) < C_ B(a_ 2) < \dots < C_ B(a_ r)\) with \(a_ 1,\dots,a_ r \in A\) satisfies \(r \leq n\). Then \(B/C_ B(p^{n-1}A)\) has Prüfer rank at most \(n -1\). This result is applied to prove that if \(G\) is a soluble stable group having a nilpotent normal subgroup \(N_ 1\) such that \(G/N_ 1\) is a torsion group then \(G\) has a nilpotent normal subgroup \(N\) such that \(G/N\) is an abelian-by-finite torsion group of finite Prüfer rank. The case where \(N_ 1 = 1\) follows from a result of the reviewer and \textit{B. Hartley} [J. Algebra 61, 328-334 (1979; Zbl 0427.20028)].