The non-abelian tensor square of residually finite groups

Abstract

Abstract. Let m, n be positive integers and p a prime. We denote by ν(G) an extension of the non-abelian tensor square G ⊗ G by G × G. We prove that if G is a residually finite group satisfying some non-trivial identity f ≡ 1 and for every x, y ∈ G there exists a p-power q = q(x, y) such that [x,′ is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every x, y ∈ G there exists a p-power q = q(x, y) dividing p m such that [x, y ϕ ] q is left n-Engel, then the non-abelian tensor square G ⊗ G is locally virtually nilpotent (Theorem B).

Non-abelian tensor product of residually finite groups

“…In the present paper and under appropriate conditions, we further extend this result to arbitrary non-abelian tensor product of groups [G, H ϕ ]. In the same paper [1] we also proved that if G is a residually finite group in which for every x, y ∈ G there exists a p-power q…”

“…In the article [1] the authors generalize in a certain way Shumyatsky's result from G to [G, G ϕ ] and ν(G) , by proving that, given a prime p and a residually finite group G satisfying some non-trivial identity, if every tensor has p-power order then [G, G ϕ ] and ν(G) are locally finite. In the present paper and under appropriate conditions, we further extend this result to arbitrary non-abelian tensor product of groups [G, H ϕ ].…”

Finiteness of homotopy groups related to the non-abelian tensor product

“…Many authors have studied some finiteness conditions for the non-abelian tensor product of groups (cf. [2,4,11,12,15,18]). For instance, in [15], Moravec proved that if G, H are locally finite groups acting compatibly on each other, then so is G ⊗ H .…”

“…When G = H and all actions are by conjugation, we simply write T ⊗ (G) instead of T ⊗ (G, G). A number of structural results for the non-abelian tensor product of groups (and related constructions) in terms of the set of tensors were presented in [2][3][4][5]21].…”

Finiteness of homotopy groups related to the non-abelian tensor product

“…When G = H and all actions are by conjugation, we simply write T ⊗ (G) instead of T ⊗ (G, G). A number of structural results for the non-abelian tensor product of groups (and related constructions) in terms of the set of tensors where presented in [2,3,4,5,21]).…”