Paper accepted for publication

IJAC-coverThe 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 g be an element of a group G. For a positive integer n, let E_n(g) be the subgroup generated by all commutators [...[[x,g],g],\dots, g] over x\in G, where g is repeated n times. We prove that if G is a profinite group such that for every g\in G there is n=n(g) such that E_n(g) is finite, then G has a finite normal subgroup N such that G/N 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 G, we prove that if, for some n, |E_n(g)|\leq m for all g\in G, then the order of the nilpotent residual \gamma _{\infty} (G) is bounded in terms of m.


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