On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks

2020/10/16 by E. I. Khukhro, Khukhro, E. I., Pavel Shumyatsky +1
Mathematics · Medicine · Computer Science · #Finite Group Theory Research #Chronic Myeloid Leukemia Treatments #Cooperative Communication and Network Coding

paper · pdf · doi:10.48550/arxiv.2010.08616

Abstract

A left Engel sink of an element g of a group G is a set \mathscr E(g) such that for every x∈ G all sufficiently long commutators [...[[x,g],g],… ,g] belong to \mathscr E(g). (Thus, g is a left Engel element precisely when we can choose \mathscr E(g)=\ 1\.) We prove that if a finite group G admits an automorphism φ of prime order coprime to |G| such that for some positive integer m every element of the centralizer C_G(φ) has a left Engel sink of cardinality at most m, then the index of the second Fitting subgroup F₂(G) is bounded in terms of m. A right Engel sink of an element g of a group G is a set \mathscr R(g) such that for every x∈ G all sufficiently long commutators [...[[g,x],x],… ,x] belong to \mathscr R(g). (Thus, g is a right Engel element precisely when we can choose \mathscr R(g)=\ 1\.) We prove that if a finite group G admits an automorphism φ of prime order coprime to |G| such that for some positive integer m every element of the centralizer C_G(φ) has a right Engel sink of cardinality at most m, then the index of the Fitting subgroup F₁(G) is bounded in terms of m.

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• Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks
• On profinite groups with automorphisms whose fixed points have countable Engel sinks
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