School of Mathematics & Physics, University of Lincoln
E. I. Khukhro and P. Shumyatsky, Almost Engel finite and profinite groups, submitted, 2015; arXiv:1512.06097.
Abstract: Let be an element of a group . For a positive integer , let be the subgroup generated by all commutators over , where is repeated times. We prove that if is a profinite group such that for every there is such that is finite, then has a finite normal subgroup such that is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group , we prove that if, for some , for all , then the order of the nilpotent residual is bounded in terms of .