A survey of non-abelian tensor products of groups and related constructions

Nakaoka, Irene N.; Rocco, Noraí R.

doi:10.5269/bspm.v30i1.13350

Cited by 6 publications

(7 citation statements)

References 26 publications

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“…The motivation for studying ν(G) is the commutator connection: indeed, the map Φ : [15,Proposition 2.6]. Therefore, from now on we shall identify the non-abelian tensor square G ⊗ G with the subgroup [G, G ϕ ] of ν(G) and write [g, h ϕ ] instead of g ⊗ h, for all g, h ∈ G. For a fuller treatment we refer the reader to [8,12].…”

Section: Introductionmentioning

confidence: 99%

Non-abelian tensor square and related constructions of $p$-groups

Bastos,

de Melo,

Gonçalves

et al. 2019

Preprint

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Let G be a group. We denote by ν(G) a certain extension of the non-abelian tensor square [G, G ϕ ] by G × G. We prove that if G is a finite potent p-group, then [G, G ϕ ] and the k-th term of the lower central series γ k (ν(G)) are potently embedded in ν(G) (Theorem A). Moreover, we show that if G is a potent p-group, then the exponent exp(ν(G)) divides p • exp(G) (Theorem B). We also study the weak commutativity construction of powerful p-groups (Theorem C).

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Section: Introductionmentioning

confidence: 99%

Non-abelian tensor square and related constructions of $p$-groups

Bastos,

de Melo,

Gonçalves

et al. 2019

Preprint

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“…The study of the non-abelian tensor square of groups from a group theoretic point of view was initiated by R. Brown, D. L. Johnson and E. F. Robertson [5]. For a deeper discussion of the non-abelian tensor square and related constructions we refer the reader to [10,11] (see also [4]).…”

Section: Introductionmentioning

confidence: 99%

Boundedly finite conjugacy classes of tensors

Bastos,

Monetta

2019

Preprint

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Let n be a positive integer and let G be a group. We denote by ν(G) a certain extension of the non-abelian tensor squareWe prove that if the size of the conjugacy class x ν(G) ≤ n for every x ∈ T ⊗ (G), then the second derived subgroup ν(G) ′′ is finite with n-bounded order. Moreover, we obtain a sufficient condition for a group to be a BFC-group.2010 Mathematics Subject Classification. 20E34, 20J06. Key words and phrases. Structure theorems; Finiteness conditions; Non-abelian tensor square of groups; BFC-groups.

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“…where the dots mean (internal) semidirect products. For a deeper discussion of non-abelian tensor product and related constructions we refer the reader to [8,13].…”

Section: Introductionmentioning

confidence: 99%

The order of the non-abelian tensor product of groups

Bastos,

Nakaoka,

Rocco

2018

Preprint

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Let G and H be groups that act compatibly on each other. We denote by [G, H] the derivative subgroup of G under H. We prove that if the set {g −1 g h | g ∈ G, h ∈ H} has m elements, then the derivative [G, H] is finite with m-bounded order. Moreover, we show that if the set of all tensors T ⊗ (G, H) = {g ⊗ h | g ∈ G, h ∈ H} has m elements, then the non-abelian tensor product G ⊗ H is finite with m-bounded order. We also examine some finiteness conditions for the non-abelian tensor square of groups.

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A survey of non-abelian tensor products of groups and related constructions

Abstract: We report on a group construction in connection with non-abelian tensor products of groups and recent development in non-abelian tensor products and q-tensor products.

Cited by 6 publications

References 26 publications

Non-abelian tensor square and related constructions of $p$-groups

Non-abelian tensor square and related constructions of $p$-groups

Boundedly finite conjugacy classes of tensors

The order of the non-abelian tensor product of groups

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