2016/10/06 by E. I. Khukhro, Khukhro, E. I., Pavel Shumyatsky +1
Mathematics · Medicine · #Finite Group Theory Research #Chronic Myeloid Leukemia Treatments
paper · pdf · doi:10.48550/arxiv.1610.02079
We say that a group G is almost Engel if for every g∈ G there is a finite set \mathscr E(g) such that for every x∈ G all sufficiently long commutators [...[[x,g],g],… ,g] belong to \mathscr E(g), that is, for every x∈ G there is a positive integer n(x,g) such that [...[[x,g],g],… ,g]∈ \mathscr E(g) if g is repeated at least n(x,g) times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose \mathscr E(g)=\ 1\ for all g∈ G.) We prove that if a compact (Hausdorff) group G is almost 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 E(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.