Abstract. The operator of weak commutativity between isomorphic groups H and H ψ was defined by Sidki asIt 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 ⊗ 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 ⊗ H is polycyclic if H is polycyclic.

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The operator χ of weak commutativity between isomorphic groupsis known to preserve group properties such as finiteness, solvability and polycyclicity. We introduce here the group constructionThe group