Engel-like conditions in fixed points of automorphisms of profinite groups

Acciarri, Cristina; Silveira, Danilo

doi:10.1007/s10231-019-00872-7

Cited by 6 publications

(3 citation statements)

References 24 publications

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“…Some of the best-known examples include Thompson's theorem [23] on the nilpotency of a finite group with a fixedpoint-free automorphism of prime order, as well as numerous other papers on finite groups admitting automorphisms with various restrictions on their fixed points. This type of results in relation to automorphisms whose fixed points have restrictions on their Engel sinks were recently obtained in [1,2,3,4,5,20].…”

Section: Introductionmentioning

confidence: 68%

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

Khukhro

Shumyatsky

2020

Preprint

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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}.) 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 F 2 (G) is bounded in terms of m.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}.) 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 F 1 (G) is bounded in terms of m.

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Section: Introductionmentioning

confidence: 68%

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

Khukhro

Shumyatsky

2020

Preprint

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“…Results of this kind have also been recently extended to profinite groups admitting a non-cyclic abelian finite group acting by coprime automorphisms [2,3,4,5,6]. In particular, Acciarri, Shumyatsky, and Silveira [3] proved the following.…”

Section: Introductionmentioning

confidence: 95%

On profinite groups with automorphisms whose fixed points have countable Engel sinks

Khukhro

Shumyatsky

2021

Isr. J. Math.

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An 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 an Engel element precisely when we can choose E (g) = {1}.) It is proved that if a profinite group G admits an elementary abelian group of automorphisms A of coprime order q 2 for a prime q such that for each a ∈ A \ {1} every element of the centralizer C G (a) has a countable (or finite) Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

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“…Other results concerning automorphisms whose fixed points satisfy restrictions on their Engel sinks were obtained in [16,17,18,19,20,21]. Generalizations of Engel conditions for finite, profinite, and compact groups using the concept of Engel sinks were considered in [22,23,24,25,26,28].…”

Section: Introductionmentioning

confidence: 99%

Finite groups with a soluble group of coprime automorphisms whose fixed points have bounded Engel sinks

Khukhro¹,

Shumyatsky²

2023

Preprint

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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 C G (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 C G (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 4 α(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}.

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Engel-like conditions in fixed points of automorphisms of profinite groups

Cited by 6 publications

References 24 publications

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

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

On profinite groups with automorphisms whose fixed points have countable Engel sinks

Finite groups with a soluble group of coprime automorphisms whose fixed points have bounded Engel sinks

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