Finite normal subgroups of strongly verbally closed groups

TL;DR

This paper generalizes a known property of finite strongly verbally closed groups to finite normal subgroups, revealing structural limitations of certain finitely generated nilpotent groups.

Contribution

It extends the result about the center of finite strongly verbally closed groups to finite normal subgroups in general, and identifies classes of nilpotent groups that are not strongly verbally closed.

Findings

Finite normal subgroups in strongly verbally closed groups have a specific structure.

Finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed.

The paper generalizes previous results about the center of such groups.

Abstract

In the recent paper by A. A. Klyachko, V. Yu. Miroshnichenko, and A. Yu. Olshanskii, it is proven that the center of any finite strongly verbally closed group is its direct factor. One of the results of the current paper is the generalization of this nontrivial fact to the case of finite normal subgroups of any strongly verbally closed groups. It follows from this generalization that finitely generated nilpotent groups with nonabelian torsion subgroups are not strongly verbally closed.