Almost Engel finite and profinite groups

2015/12/18 by E. I. Khukhro, Khukhro, E. I., Pavel Shumyatsky +1
Mathematics · Engineering · Medicine · #Finite Group Theory Research #graph theory and CDMA systems #Chronic Myeloid Leukemia Treatments

paper · doi:10.48550/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],… ,g] over x∈ G, where g is repeated n times. We prove that if G is a profinite group such that for every g∈ 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)|≤ m for all g∈ G, then the order of the nilpotent residual γ_∞(G) is bounded in terms of m.

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• Finite and profinite groups with small Engel sinks of p-elements

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• Almost Engel compact groups
• Compact groups all elements of which are almost right Engel
• On profinite groups with automorphisms whose fixed points have countable Engel sinks
• Finite groups with Engel sinks of bounded rank
• Compact groups with countable Engel sinks