Weak commutativity between two isomorphic polycyclic groups
-
Bruno César R. Lima
and
Ricardo N. Oliveira
Abstract
An operator of weak commutativity between isomorphic groups, H and Hψ, was defined by Sidki as χ(H) = 〈HHψ | [h,hψ] = 1 for all h ∈ H〉, where ψ : h ↦ hψ 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.
Funding source: PROCAD-CAPES
We are grateful to Professor S. Sidki for suggesting this problem and for his support and encouragement.
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Mapping tori of free group automorphisms, and the Bieri–Neumann–Strebel invariant of graphs of groups
- Residual properties of graph products of groups
- On hereditarily just infinite profinite groups obtained via iterated wreath products
- Weak commutativity between two isomorphic polycyclic groups
- Isomorphisms and automorphisms of extensions of bilinear dimensional dual hyperovals and quadratic APN functions
- Finite groups in which pronormality and 𝔉-pronormality coincide
- Towards Thompson's conjecture for alternating and symmetric groups
- Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups
- A criterion for a finite permutation group to be transitive
- Inverse Glauberman–Isaacs correspondence and subnormal subgroups
Articles in the same Issue
- Frontmatter
- Mapping tori of free group automorphisms, and the Bieri–Neumann–Strebel invariant of graphs of groups
- Residual properties of graph products of groups
- On hereditarily just infinite profinite groups obtained via iterated wreath products
- Weak commutativity between two isomorphic polycyclic groups
- Isomorphisms and automorphisms of extensions of bilinear dimensional dual hyperovals and quadratic APN functions
- Finite groups in which pronormality and 𝔉-pronormality coincide
- Towards Thompson's conjecture for alternating and symmetric groups
- Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups
- A criterion for a finite permutation group to be transitive
- Inverse Glauberman–Isaacs correspondence and subnormal subgroups
