Finite normal subgroups of strongly verbally closed groups (Q6611933)

Let \(G\) be a group. A subgroup \(H \leq G\) is verbally closed if any equation of the form \(w(x_{1}, x_{2}, \ldots, x_{n})=h\) where \(w\) is an element of the free group \(F(x_{1},x_{2}, \ldots, x_{n})\) and \(h \in H\), having solutions in \(G\), has a solution in \(H\). If each system of equations with coefficients from \(H\), \(\{w_{1}=1,w_{2}=1, \ldots, w_{m}=1\}\), where \(w_{i} \in H \ast F(x_{1}, x_{2}, \ldots x_{n})\), having solutions in \(G\) has a solution in \(H\), then the subgroup \(H\) is said to be algebraically closed in \(G\). A group \(G\) is called strongly verbally closed if it is algebraically closed in any group containing \(G\) as a verbally closed subgroup (so verbal closedness is a property of an abstract group).\N\N\textit{A. A. Klyachko} et al. [J. Algebra Appl. 22, No. 9, Article ID 2350188, 19 p. (2023; Zbl 1521.20102)] proved that the center of any finite strongly verbally closed group is a direct factor. In the paper under review, the author extends this result to the case of finite normal subgroups of any strongly verbally closed group. So he is able to prove Theorem 1.1: Let \(G\) be a finitely generated nilpotent group with non-abelian torsion subgroup. Then \(G\) is not strongly verbally closed.\N\NA property that is stronger than the strong verbal closedness is the property of being a strong retract. A group \(H\) is called a strong retract if it is a retract of any group \(G \geq H\) from the variety generated by the group \(H\). The second main result of this paper is Theorem 1.2: Let \(G\) be a nilpotent strong retract. Then \(G\) is abelian. Moreover, \(G\) is either divisible or it has bounded period and its decomposition as a direct sum of primary cyclic factors has the property that the orders of any distinct summands are either equal or coprime.