Weak Commutativity Between Two Isomorphic Polycyclic Groups

2014/09/19 by Lima, Bruno César Rodrigues, de Oliveira, Ricardo Nunes · 1 citation
#20F99 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1409.5511

Abstract

The operator of weak commutativity between isomorphic groups H and H^ψ was defined by Sidki as χ(H)=⟨ H H^ψ| \lbrack h,h^ψ]=1 ∀ h∈ H⟩ .% 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⊗ 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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