Residually finite subgroups of some countable McLain groups. (Q1047898)

The authors prove that certain torsion-free McLain groups have subgroups of periodic index that are residually finite. It was proven [\textit{M. R. Dixon, M. J. Evans} and \textit{H. Smith}, J. Algebra 320, No. 1, 81-85 (2008; Zbl 1160.20023)] that a countable torsion-free locally nilpotent group \(G\) that is also soluble has a residually finite subgroup \(J\) of periodic index in \(G\), that is, whose isolator in \(G\) is \(G\) itself. It also has been proven [in \textit{M. R. Dixon, M. J. Evans} and \textit{H. Smith}, J. Group Theory 9, No. 3, 383-395 (2006; Zbl 1120.20030)] that any countable group \(G\) that embeds in a simple group that is locally (soluble-by-finite) must at least have a residually finite subgroup of periodic index. The authors of the article under review describe the following important property of McLain groups. Theorem. Let \(A\) be a commutative ring with identity whose additive group has finite torsion-free rank and let \(R\) be a ring embeddable in a polynomial ring on a (possibly infinite) set of commuting indeterminates over \(A\). Then the McLain group \(M(\mathsf Q,R)\) has a residually finite subgroup of periodic index.