Centralizer extension in nilpotent groups. (Q359321)

The irreducible coordinate groups of algebraic sets for nilpotent \(R\)-groups (where \(R\) is a Euclidean ring) of class 2 are studied. Let \(G\) be a group and let \(C=C_G(g)\) be the centralizer of a nontrivial element \(g\) in \(G\). Then the group \(H=\langle G,t\mid t^{-1}ct=c\text{ for all }c\in C_G(x)\rangle_{\text{var}(G)}\) is said to be the centralizer extension of \(G\) of rank 1. A series of centralizer extensions \(G=G_1\leq\cdots\leq G_k=H\) defines the iterated centralizer extension \(H\) of \(G\). It is proved that an analog of Lyndon's theorem for free groups is true in the case of nilpotent \(R\)-torsion-free \(R\)-groups of class 2 for every Euclidean ring \(R\). An analog of Kharlampovich-Myasnikov's theorem is not true in this case.