On Nonabelian Tensor Analogues of 2-Engel Conditions

Abstract

Abstract. Tensor analogues of right 2-Engel elements in groups were introduced by D. P. Biddle and L.-C. Kappe. We investigate the properties of right 2-Engel tensor elements and introduce the concept of 2 ⊗ -Engel margin. With the help of these results we describe the structure of 2 ⊗ -Engel groups. In particular, we prove a tensor version of Levi's theorem for 2-Engel groups and determine tensor squares of two-generator 2 ⊗ -Engel p-groups.2000 Mathematics Subject Classification. 20F45, 20F99.

Non-Abelian Tensor Analogues of 2-Auto Engel Groups

“…Moravec in [9] shows that if G is a 2 ⊗ -Engel group, then the non-abelian tensor product G ⊗ G is abelian and C ⊗ G (g) is a normal subgroup of G, for each g ∈ G. We say a group G is 2 ⊗ -auto Engel group if [g, α] ⊗ α = 1 ⊗ , for all g ∈ G and α ∈ Aut(G).…”

“…see also [1,9]. The set of right n ⊗ -Engel elements has been studied by several authors (see [1,9,11,12]).…”

“…The set of right n ⊗ -Engel elements has been studied by several authors (see [1,9,11,12]). Biddle and Kappe [1] proved that R ⊗ 2 (G) is always a characteristic subgroup of G contained in R 2 (G).…”

“…Biddle and Kappe [1] proved that R ⊗ 2 (G) is always a characteristic subgroup of G contained in R 2 (G). In [9], Moravec determined some further information on R ⊗ 2 (G) and defined the concept of 2 ⊗ -Engel groups, which reads as G is a 2 ⊗ -Engel group when [g, h] ⊗ h = 1 ⊗ for all g, h ∈ G. He also showed that if G is a 2 ⊗ -Engel group, then the non-abelian tensor square G ⊗ G is abelian.…”

“…The concept of tensor analogues of right 2-Engel elements in groups were defined and studied by Biddle and Kappe [1] and Moravec [9]. Using the automorphisms of a given group G, we introduce the notion of tensor analogue of 2-auto Engel elements in G and investigate their properties.…”