Weak Commutativity Between Two Isomorphic Polycyclic Groups

Abstract

The operator of weak commutativity between isomorphic groups $H$ and $H^{ψ}$ was defined by Sidki as \begin{equation*} χ(H)=\left\langle H\,H^{ψ}\mid \lbrack h,h^{ψ}]=1\,\forall \,h\in H\right\rangle \text{.} \end{equation*}% It is known that the operator $χ$ preserves group properties such as finiteness, solubility and also nilpotency for finitely generated groups. We prove in this work that $χ$ 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.