scholarly journals WORDS AND PRONILPOTENT SUBGROUPS IN PROFINITE GROUPS

2014 ◽  
Vol 97 (3) ◽  
pp. 343-364 ◽  
Author(s):  
E. I. KHUKHRO ◽  
P. SHUMYATSKY
Keyword(s):  
Profinite Group ◽  
Profinite Groups ◽  
Locally Finite ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Commutator Word

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

scholarly journals Profinite groups with restricted centralizers of commutators

2019 ◽  
Vol 150 (5) ◽  
pp. 2301-2321 ◽  
Author(s):  
Eloisa Detomi ◽  
Marta Morigi ◽  
Pavel Shumyatsky
Keyword(s):  
Finite Index ◽  
Profinite Group ◽  
Profinite Groups ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Commutator Word

AbstractA group G has restricted centralizers if for each g in G the centralizer $C_G(g)$ either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present paper we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.

scholarly journals ON GROUPS ADMITTING A WORD WHOSE VALUES ARE ENGEL

2013 ◽  
Vol 23 (01) ◽  
pp. 81-89 ◽  
Author(s):  
RAIMUNDO BASTOS ◽  
PAVEL SHUMYATSKY ◽  
ANTONIO TORTORA ◽  
MARIA TOTA
Keyword(s):  
Finite Group ◽  
Multilinear Commutator ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Verbal Subgroup ◽  
Locally Nilpotent ◽  
Locally Graded Group ◽  
Commutator Word ◽  
Positive Integers

Let m, n be positive integers, v a multilinear commutator word and w = vm. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.

scholarly journals ON THE EXPONENT OF A VERBAL SUBGROUP IN A FINITE GROUP

2012 ◽  
Vol 93 (3) ◽  
pp. 325-332 ◽  
Author(s):  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Nilpotent Subgroup ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
A Finite Group ◽  
Commutator Word

AbstractLet $w$ be a multilinear commutator word. We prove that if $e$ is a positive integer and $G$ is a finite group in which any nilpotent subgroup generated by $w$-values has exponent dividing $e$, then the exponent of the corresponding verbal subgroup $w(G)$ is bounded in terms of $e$ and $w$only.

scholarly journals DERIVED SUBGROUPS OF FIXED POINTS IN PROFINITE GROUPS

2011 ◽  
Vol 54 (1) ◽  
pp. 97-105
Author(s):  
CRISTINA ACCIARRI ◽  
ALINE DE SOUZA LIMA ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Abelian Group ◽  
Fixed Points ◽  
Profinite Group ◽  
Elementary Abelian Group ◽  
Profinite Groups ◽  
Locally Finite ◽  
Group Of Automorphisms

AbstractThe main result of this paper is the following theorem. Let q be a prime and A be an elementary abelian group of order q3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that CG(a)′ is periodic for each a ∈ A#. Then G′ is locally finite.

scholarly journals On locally graded groups with a word whose values are Engel

2015 ◽  
Vol 59 (2) ◽  
pp. 533-539 ◽  
Author(s):  
Pavel Shumyatsky ◽  
Antonio Tortora ◽  
Maria Tota
Keyword(s):  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Locally Nilpotent ◽  
Locally Graded Group ◽  
Commutator Word ◽  
Locally Graded Groups ◽  
Positive Integers

AbstractLet m, n be positive integers, let υ be a multilinear commutator word and let w = υm. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.

scholarly journals ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

2021 ◽  
pp. 1-15
Author(s):  
ELOISA DETOMI ◽  
MARTA MORIGI ◽  
PAVEL SHUMYATSKY
Keyword(s):  
Finite Group ◽  
Soluble Group ◽  
Nilpotent Subgroup ◽  
Finite Soluble Group ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
A Finite Group ◽  
Commutator Word

Abstract We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

scholarly journals On finite groups in which commutators are covered by Engel subgroups

2019 ◽  
Vol 22 (6) ◽  
pp. 1049-1057
Author(s):  
Pavel Shumyatsky ◽  
Danilo Silveira
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Multilinear Commutator ◽  
Verbal Subgroup ◽  
Bounded Number ◽  
A Finite Group ◽  
Commutator Word ◽  
Positive Integers

Abstract Let {m,n} be positive integers and w a multilinear commutator word. Assume that G is a finite group having subgroups {G_{1},\ldots,G_{m}} whose union contains all w-values in G. Assume further that all elements of the subgroups {G_{1},\ldots,G_{m}} are n-Engel in G. It is shown that the verbal subgroup {w(G)} is s-Engel for some {\{m,n,w\}} -bounded number s.

scholarly journals On profinite groups in which commutators are Engel

2001 ◽  
Vol 70 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Pavel Shumyatsky
Keyword(s):  
Finite Order ◽  
Profinite Group ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Additional Assumptions

AbstractWe show that if G is a finitely generated profinite group such that [x1, x2, …, xk] is Engel for any x1, x2, …, xk ∈ G, then γ(G) is locally nilpotent, and if [x1, x2, …, xk] has finite order for any x1, x2, …, xk ∈ G then, under some additional assumptions, γk(G) is locally finite.

scholarly journals Almost Engel finite and profinite groups

2016 ◽  
Vol 26 (05) ◽  
pp. 973-983 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Profinite Group ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Nilpotent Residual ◽  
A Finite Group

Let [Formula: see text] be an element of a group [Formula: see text]. For a positive integer [Formula: see text], let [Formula: see text] be the subgroup generated by all commutators [Formula: see text] over [Formula: see text], where [Formula: see text] is repeated [Formula: see text] times. We prove that if [Formula: see text] is a profinite group such that for every [Formula: see text] there is [Formula: see text] such that [Formula: see text] is finite, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group [Formula: see text], we prove that if, for some [Formula: see text], [Formula: see text] for all [Formula: see text], then the order of the nilpotent residual [Formula: see text] is bounded in terms of [Formula: see text].

On Nonabelian H2 for Profinite Groups

1992 ◽  
Vol 44 (2) ◽  
pp. 388-399
Author(s):  
K.-H. Ulbrich
Keyword(s):  
Normal Subgroup ◽  
Exact Sequence ◽  
Profinite Group ◽  
Profinite Groups ◽  
Image Position

Let G be a profinite group. We define an extension (E, J) of G by a group A to consist of an exact sequence of groups together with a section j : G → E of K satisfying: for some open normal subgroup Sof G, and the map is continuous (A being discrete).

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