School of Mathematics & Physics, University of Lincoln
The paper by E. I. Khukhro and P. Shumyatsky, Almost Engel finite and profinite groups has been accepted for publication in the International Journal of Algebra and Computation, ISSN 0218-1967, see also 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 .