Khukhro, Evgeny and Shumyatsky, Pavel (2023) Finite groups with a soluble group of coprime automorphisms whose fixed points have bounded Engel sinks. Algebra and Logic . ISSN 00025232
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Abstract
Suppose that a finite group G admits a soluble group of coprime automorphisms A. We prove that if, for some positive integer m, every element of the centralizer CG(A) has a left Engel sink of cardinality at most m (or a right Engel sink of cardinality at most m), then G has a subgroup of (A,m)bounded index which has Fitting height at most 2α(A)+2, where α(A) is the composition length of A. We also prove that if, for some positive integer r, every element of the centralizer CG(A) has a left Engel sink of rank at most r (or a right Engel sink of rank at most r), then G has a subgroup of (A,r)bounded index which has Fitting height at most 4α(A)+4α(A)+3. Here, a left Engel sink of an element g of a group G is a set E(g) such that for every x∈G all sufficiently long commutators [...[[x,g],g],…,g] belong to E(g). (Thus, g is a left Engel element precisely when we can choose E(g)={1}.) A right Engel sink of an element g of a group G is a set R(g) such that for every x∈G all sufficiently long commutators [...[[g,x],x],…,x] belong to R(g). (Thus, g is a right Engel element precisely when we can choose R(g)={1}.)
Keywords:  Finite groups, Engel condition, Fitting subgroup, Fitting height, Automorphism 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  COLLEGE OF HEALTH AND SCIENCE > School of Mathematics and Physics 
ID Code:  54950 
Deposited On:  09 Jun 2023 14:46 
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