On Non-Abelian Tensor Analogues of 3-Engel and 4-Engel Groups

Abstract

A group G is said to be an n _⊗-Engel, if [y, _n−1 x] ⊗ x = 1 for all x, y ∈ G, and we say a group G is tensor nilpotent of class at most n, if . In this article, we show that if G is a 3_⊗-Engel group, then ⟨ x, x ^y  ⟩ is tensor nilpotent of class at most 2, for all x, y ∈ G. We also prove that if G is a 4_⊗-Engel group and G ⊗ G is torsion-free, then ⟨ x, x ^y  ⟩ is tensor nilpotent of class at most 4, for all x, y ∈ G.

Key Words:

• Engel group
• Nilpotent group
• Non-abelian tensor square

2010 Mathematics Subject Classification:

• 20F45
• 20F99

Notes

Communicated by A. Olshanskii.