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

• Published in: Communications in algebra Vol. 43; no. 10; pp. 4415 - 4421
• Main Authors: Hokmabadi, Azam, Mohammadzadeh, Elaheh, Golmakani, Hanieh, Mohammadzadeh, Fahimeh
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 03.10.2015
• Subjects: Engel group; Nilpotent group; Non-abelian tensor square
• ISSN: 0092-7872; 1532-4125
• DOI: 10.1080/00927872.2014.946140

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.