Weak Commutativity Between Two Isomorphic Polycyclic Groups

TL;DR

This paper investigates the operator of weak commutativity between isomorphic polycyclic groups, proving it preserves polycyclic properties and extending known results about the non-abelian tensor square of such groups.

Contribution

It demonstrates that the weak commutativity operator preserves polycyclic and polycyclic-by-finite properties, extending understanding of these properties in group theory.

Findings

The operator $ ext{χ}$ preserves polycyclic and polycyclic-by-finite properties.

The non-abelian tensor square $H ensor H$ preserves polycyclic-by-finite property.

Extension of previous results on polycyclic groups and their tensor squares.

Abstract

The operator of weak commutativity between isomorphic groups $H$ and $H^{\psi }$ was defined by Sidki as \begin{equation*} \chi (H)=\left\langle H\,H^{\psi }\mid \lbrack h,h^{\psi }]=1\,\forall \,h\in H\right\rangle \text{.} \end{equation*}% It is known that the operator $\chi $ preserves group properties such as finiteness, solubility and also nilpotency for finitely generated groups. We prove in this work that $\chi $ preserves the properties of being polycyclic and polycyclic by finite. As a consequence of this result, we conclude that the non-abelian tensor square $H\otimes H$ of a group $H$, defined by Brown and Loday, preserves the property polycyclic by finite. This last result extends that of Blyth and Morse who proved that $H\otimes H$ is polycyclic if $H$ is polycyclic.