The non-abelian tensor square of residually finite groups
Tóm tắt
Let m, n be positive integers and p a prime. We denote by
$$\nu (G)$$
an extension of the non-abelian tensor square
$$G \otimes G$$
by
$$G \times G$$
. We prove that if G is a residually finite group satisfying some non-trivial identity
$$f \equiv ~1$$
and for every
$$x,y \in G$$
there exists a p-power
$$q=q(x,y)$$
such that
$$[x,y^{\varphi }]^q = 1$$
, then the derived subgroup
$$\nu (G)'$$
is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every
$$x,y \in G$$
there exists a p-power
$$q=q(x,y)$$
dividing
$$p^m$$
such that
$$[x,y^{\varphi }]^q$$
is left n-Engel, then the non-abelian tensor square
$$G \otimes G$$
is locally virtually nilpotent (Theorem B).
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