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

2025/05/02 by Lucas Dal Berto, Berto, Lucas Dal, Jhone Caldeira +3
Computer Science · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2505.01409

openalex publication_date 2025/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A (left) Engel sink of an element g of a group G is a subset containing all sufficiently long commutators [...[[x,g],g],...,g], where x ranges over G. We prove that if p is a prime and G a finite group in which, for some positive integer m, every p-element has an Engel sink of cardinality at most m, then G has a normal subgroup N such that G/N is a p'-group and the index [N:O_p(G)] is bounded in terms of m only. Furthermore, if G is a profinite group in which every p-element possesses a finite Engel sink, then G has a normal subgroup N such that N is virtually pro-p while G/N is a pro-p' group.

Citations

• Almost Engel compact groups
• Almost Engel finite and profinite groups
• Engel-type subgroups and length parameters of finite groups
• Nonsoluble and non-p-soluble length of finite groups

Related

• Finite groups with Engel sinks of bounded rank
• On finite groups with an automorphism of prime order whose fixed points have bounded Engel sinks
• On groups in which Engel sinks are cyclic
• On profinite groups with automorphisms whose fixed points have countable Engel sinks
• Compact groups with countable Engel sinks