Weak commutativity between two isomorphic polycyclic groups

Abstract

Abstract An operator of weak commutativity between isomorphic groups, H and H ψ, was defined by Sidki as χ(H) = 〈H H ψ | [h,h ψ] = 1 for all h ∈ H〉, where ψ : h ↦ h ψ for all h ∈ H defines an isomorphism. 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 or polycyclic-by-finite. As a consequence of this result, we conclude that the non-abelian tensor square H ⊗ H of a group H as defined by Brown and Loday preserves the property of being polycyclic-by-finite. This last result extends work of Blyth and Morse who proved that H ⊗ H is polycyclic if H is polycyclic.