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Finite normal subgroups of strongly verbally closed groups

Filipp D. Denissov

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.

Finite normal subgroups of strongly verbally closed 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.
Paper Structure (4 sections, 12 theorems, 37 equations)

This paper contains 4 sections, 12 theorems, 37 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Nilpotent strong retracts
  3. 3. Finite normal subgroups of strongly verbally closed groups
  4. 4. Nilpotent non-strongly-verbally-closed groups

Key Result

Proposition 1

An abelian group $G$ of unbounded period is a strong retract if and only if it is divisible.

Theorems & Definitions (24)

  • Proposition 1
  • Proof
  • Proposition 2
  • Proof
  • Proposition 3
  • Proof
  • Proposition 4
  • Proof
  • Lemma 1
  • Proof
  • ...and 14 more