2018/07/14 by E. I. Khukhro, Khukhro, E. I., Pavel Shumyatsky +1
Computer Science · Mathematics · #20D25 #20E18 #20F45 #Coding theory and cryptography #Cooperative Communication and Network Coding #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1807.06452
openalex publication_date 2018/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We say that an element g of a group G is almost right Engel if there is a finite set \mathscr R(g) such that for every x∈ G all sufficiently long commutators [...[[g,x],x],… ,x] belong to \mathscr R(g), that is, for every x∈ G there is a positive integer n(x,g) such that [...[[g,x],x],… ,x]∈ \mathscr R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose \mathscr R(g)=\ 1\. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound |\mathscr R(g)|≤ m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent and previous results of the authors about compact groups all elements of which are almost left Engel.