On Pro-p Groups Admitting a Fixed-Point-Free Automorphism

Shumyatsky, Pavel

doi:10.1006/jabr.1999.8267

Cited by 6 publications

(4 citation statements)

References 11 publications

(18 reference statements)

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“…Then [a, y 2 n(c−1) ] = 1. (6.2) This follows from well-known commutator formulae (and for any p-group); see, for example, [21,Lemma 4.1].…”

Section: Bounding the Nonprosoluble Lengthmentioning

confidence: 87%

Compact groups with countable Engel sinks

Khukhro

Shumyatsky

2019

Preprint

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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 every element of a compact (Hausdorff) group G has a countable (or finite) Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent. This settles a question suggested by J. S. Wilson.

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“…Then [a, y 2 n(c−1) ] = 1. (6.2) This follows from well-known commutator formulae (and for any p-group); see, for example, [21,Lemma 4.1].…”

Section: Bounding the Nonprosoluble Lengthmentioning

confidence: 87%

Compact groups with countable Engel sinks

Khukhro

Shumyatsky

2019

Preprint

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“…Some other special situations. For residually finite groups, analogues of Kreknin's theorem under some additional conditions were proved by N. Rocco and P. Shumyatsky [95], A. Shalev [102], and P. Shum~-atsky [110,111]. These works use the deep theorem of E. I. Zelmanov [129] on nilpotency of Lie algebras satisfying polynomial identities and generated by finitely many elements, in which all commutators are adnilpotent.…”

Section: Proposition (A) For a ~O-invariant Subgroup H We Have (Re(mentioning

confidence: 96%

Almost regular automorphisms

Khukhro¹

1999

J Math Sci

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“…This follows from well-known commutator formulae (and for any p-group); see, for example,[20, Lemma 4.1].In particular, for any z ∈ N by using (6.2) we obtain[z, c y 2 m(c−1) ] = [az, c y 2 m(c−1) ] = 1,(6.3)since az, y 2 m(c−1) is a subgroup of az, y , which is nilpotent of class c by (6.1).Our aim is to show that there is a uniform bound, in terms of |G : K|, c, and m, for the nonsoluble length of all finite quotients of G by open normal subgroups. Let M be an open normal subgroup of G and let the bar denote the images in Ḡ = G/M.…”

mentioning

confidence: 88%

Strong conciseness of Engel words in profinite groups

Khukhro¹,

Shumyatsky²

2021

Preprint

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A group word w is said to be strongly concise in a class C of profinite groups if, for any group G in C , either w takes at least continuum values in G or the verbal subgroup w(G) is finite. It is conjectured that all words are strongly concise in the class of all profinite groups. Earlier Detomi, Klopsch, and Shumyatsky proved this conjecture for multilinear commutator words, as well as for some other particular words. They also proved that every group word is strongly concise in the class of nilpotent profinite groups.In the present paper we prove that for any n the n-Engel word [...[x, y], y], . . . y] (where y is repeated n times) is strongly concise in the class of finitely generated profinite groups.

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On Pro-p Groups Admitting a Fixed-Point-Free Automorphism

Cited by 6 publications

References 11 publications

Compact groups with countable Engel sinks

Compact groups with countable Engel sinks

Almost regular automorphisms

Strong conciseness of Engel words in profinite groups

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