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.

On Certain Classes of Locally Soluble Groups

1962 ◽  
Vol 58 (2) ◽  
pp. 185-195
Author(s):  
J. E. Roseblade
Keyword(s):  
Finitely Generated ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Nilpotent

A group G is called locally soluble if every finitely generated subgroup of G is soluble. Terms like ‘locally nilpotent’ and ‘locally finite’ are defined similarly.

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

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].

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.

Stretchings

1996 ◽  
Vol 61 (2) ◽  
pp. 563-585 ◽  
Author(s):  
O. Finkel ◽  
J. P. Ressayre
Keyword(s):  
Finite Order ◽  
Order Type ◽  
Finitely Generated ◽  
Locally Finite ◽  
Mahlo Cardinals

AbstractA structure is locally finite if every finitely generated substructure is finite; local sentences are universal sentences all models of which are locally finite. The stretching theorem for local sentences expresses a remarkable reflection phenomenon between the finite and the infinite models of local sentences. This result in part requires strong axioms to be proved; it was studied by the second named author, in a paper of this Journal, volume 53. Here we correct and extend this paper; in particular we show that the stretching theorem implies the existence of inaccessible cardinals, and has precisely the consistency strength of Mahlo cardinals of finite order. And we present a sequel due to the first named author:(i) decidability of the spectrum Sp(φ) of a local sentence φ, below ωω; where Sp(φ) is the set of ordinals α such that φ has a model of order type α(ii) proof that bethω = sup{Sp(φ): φ local sentence with a bounded spectrum}(iii) existence of a local sentence φ such that Sp(φ) contains all infinite ordinals except the inaccessible cardinals.

Deficiency and commensurators

2018 ◽  
Vol 21 (3) ◽  
pp. 511-530
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Fundamental Group ◽  
Natural Homomorphism ◽  
Finitely Generated ◽  
Locally Finite ◽  
Finitely Generated Group ◽  
Locally Nilpotent

Abstract We show that if π is the fundamental group of a 4-dimensional infrasolvmanifold then {-2\leq\mathrm{def}(\pi)\leq 0} , and give examples realizing each value allowed by our constraints, for each possible value of the rank of {\pi/\pi^{\prime}} . We also consider the abstract commensurators of such groups. Finally, we show that if G is a finitely generated group, the kernel of the natural homomorphism from G to its abstract commensurator {\mathrm{Comm}(G)} is locally nilpotent by locally finite, and is finite if {\mathrm{def}(G)>1} .

scholarly journals Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

2021 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Prime Order ◽  
Profinite Group ◽  
Nilpotent Subgroup ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Engel Element

AbstractA right Engel sink of an element g of a group G is a set $${{\mathscr {R}}}(g)$$ R ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[g,x],x],\dots ,x]$$ [ . . . [ [ g , x ] , x ] , ⋯ , x ] belong to $${\mathscr {R}}(g)$$ R ( g ) . (Thus, g is a right Engel element precisely when we can choose $${{\mathscr {R}}}(g)=\{ 1\}$$ R ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set $${{\mathscr {E}}}(g)$$ E ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$ [ . . . [ [ x , g ] , g ] , ⋯ , g ] belong to $${{\mathscr {E}}}(g)$$ E ( g ) . (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$ E ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

scholarly journals Profinite groups in which the probabilistic zeta function has no negative coefficients

2020 ◽  
pp. 1-15
Author(s):  
Eloisa Detomi ◽  
Andrea Lucchini
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Profinite Group ◽  
Möbius Function ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Mobius Function ◽  
Formal Inverse

To a finitely generated profinite group [Formula: see text], a formal Dirichlet series [Formula: see text] is associated, where [Formula: see text] and [Formula: see text] denotes the Möbius function of the lattice of open subgroups of [Formula: see text] Its formal inverse [Formula: see text] is the probabilistic zeta function of [Formula: see text]. When [Formula: see text] is prosoluble, every coefficient of [Formula: see text] is nonnegative. In this paper we discuss the general case and we produce a non-prosoluble finitely generated group with the same property.

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.

Subgroups of free pro-p-products

1987 ◽  
Vol 101 (2) ◽  
pp. 197-206 ◽  
Author(s):  
Wolfgang Herfort ◽  
Luis Ribes
Keyword(s):  
Prime Number ◽  
Closed Subgroup ◽  
Profinite Group ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Free Products ◽  
Free Profinite Group

If F is a free profinite group, it is well known that the closed subgroups of F need not be free profinite; however, if p is a prime number, every closed subgroup of a free pro-p-group is free pio-p (cf. [2, 8, 7]). In this paper we show that there is an analogous contrast regarding the closed subgroups of free products in the category of profinite groups, and the closed subgroups of free products in the category of pro-p-groups, at least for (topologically) finitely generated subgroups.

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