scholarly journals Compact groups in which all elements have countable right Engel sinks

2020 ◽  
pp. 1-25
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Normal Subgroup ◽  
Compact Groups ◽  
Locally Nilpotent ◽  
Hausdorff Group ◽  
Engel Element

A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$ . (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$ .) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

scholarly journals Compact groups with countable Engel sinks

2020 ◽  
pp. 2050015 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Normal Subgroup ◽  
Compact Groups ◽  
Locally Nilpotent ◽  
Hausdorff Group ◽  
Engel Element

An Engel sink of an element [Formula: see text] of a group [Formula: see text] is a set [Formula: see text] such that for every [Formula: see text] all sufficiently long commutators [Formula: see text] belong to [Formula: see text]. (Thus, [Formula: see text] is an Engel element precisely when we can choose [Formula: see text].) It is proved that if every element of a compact (Hausdorff) group [Formula: see text] has a countable (or finite) Engel sink, then [Formula: see text] has a finite normal subgroup [Formula: see text] such that [Formula: see text] is locally nilpotent. This settles a question suggested by J. S. Wilson.

scholarly journals Compact groups in which all elements are almost right Engel

2019 ◽  
Vol 70 (3) ◽  
pp. 879-893 ◽  
Author(s):  
E I Khukhro ◽  
P Shumyatsky
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Compact Groups ◽  
Uniform Bound ◽  
Locally Nilpotent ◽  
Finite Set ◽  
Hausdorff Group ◽  
Engel Element

Abstract We say that an element g of a group G is almost right Engel if there is a finite set R(g) such that for every x∈G, there is a positive integer n(x,g) such that […[[g,x],x],…,x]∈R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose R(g)={1}. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound ∣R(g)∣⩽m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson–Zelmanov theorem saying that profinite Engel groups are locally nilpotent and previous results of the authors about compact groups in which all elements are almost left Engel.

scholarly journals Homomorphisms into totally disconnected, locally compact groups with dense image

2019 ◽  
Vol 31 (3) ◽  
pp. 685-701 ◽  
Author(s):  
Colin D. Reid ◽  
Phillip R. Wesolek
Keyword(s):  
Normal Subgroup ◽  
Open Subgroup ◽  
Locally Compact Groups ◽  
Group Homomorphism ◽  
Compact Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Dense Image ◽  
Profinite Completions ◽  
Totally Disconnected

Abstract Let {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.

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 LOCALLY COMPACT WREATH PRODUCTS

2018 ◽  
Vol 107 (1) ◽  
pp. 26-52 ◽  
Author(s):  
YVES CORNULIER
Keyword(s):  
Normal Subgroup ◽  
Wreath Product ◽  
Natural Extension ◽  
Wreath Products ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Intermediate Growth ◽  
Compactly Generated

Wreath products of nondiscrete locally compact groups are usually not locally compact groups, nor even topological groups. As a substitute introduce a natural extension of the wreath product construction to the setting of locally compact groups. Applying this construction, we disprove a conjecture of Trofimov, constructing compactly generated locally compact groups of intermediate growth without any open compact normal subgroup.

scholarly journals Generators for locally compact groups

1993 ◽  
Vol 36 (3) ◽  
pp. 463-467 ◽  
Author(s):  
Joan Cleary ◽  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Positive Integer ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Real Numbers ◽  
Locally Compact ◽  
Hausdorff Group

It is proved that if G is any compact connected Hausdorff group with weight w(G)≦c, ℝ is the topological group of all real numbers and n is a positive integer, then the topological group G × ℝn can be topologically generated by n + 1 elements, and no fewer elements will suffice.

On Quotients of Groups Having a Series with Finite Factors

2008 ◽  
Vol 15 (02) ◽  
pp. 337-339
Author(s):  
Chiara Nicotera
Keyword(s):  
Normal Subgroup ◽  
Locally Nilpotent

The class of groups having a series with finite factors is not quotient closed. In this note, we prove that if the group G has a series whose factors are finite and H is a locally nilpotent normal subgroup of G, then G/H also has a series whose factors are finite.

Locally compact groups: maximal compact subgroups and N-groups

1988 ◽  
Vol 104 (1) ◽  
pp. 47-64
Author(s):  
R. W. Bagley ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Compact Set ◽  
Locally Compact ◽  
Group A ◽  
Totally Disconnected ◽  
Compactly Generated

AbstractIf G is a locally compact group such thatG/G0contains a uniform compactly generated nilpotent subgroup, thenGhas a maximal compact normal subgroupKsuch thatG/Gis a Lie group. A topological groupGis an N-group if, for each neighbourhoodUof the identity and each compact setC⊂G, there is a neighbourhoodVof the identity such thatfor eachg∈G. Several results on N-groups are obtained and it is shown that a related weaker condition is equivalent to local finiteness for certain totally disconnected groups.

Compactifications of locally compact groups and quotients

1994 ◽  
Vol 116 (3) ◽  
pp. 451-463 ◽  
Author(s):  
A. T. Lau ◽  
P. Milnes ◽  
J. S. Pym
Keyword(s):  
Normal Subgroup ◽  
Compact Group ◽  
Semidirect Product ◽  
Continuous Functions ◽  
Locally Compact Group ◽  
Positive Answer ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Uniformly Continuous

AbstractLet N be a compact normal subgroup of a locally compact group G. One of our goals here is to determine when and how a given compactification Y of G/N can be realized as a quotient of the analogous compactification (ψ, X) of G by Nψ = ψ(N) ⊂ X; this is achieved in a number of cases for which we can establish that μNψ ⊂ Nψ μ for all μ ∈ X A question arises naturally, ‘Can the latter containment be proper?’ With an example, we give a positive answer to this question.The group G is an extension of N by GN and can be identified algebraically with Nx GN when this product is given the Schreier multiplication, and for our further results we assume that we can also identify G topologically with N x GN. When GN is discrete and X is the compactification of G coming from the left uniformly continuous functions, we are able to show that X is an extension of N by (GN)(X≅N x (G/N)) even when G is not a semidirect product. Examples are given to illustrate the theory, and also to show its limitations.

scholarly journals Left 3-Engel elements in groups of exponent 60

2018 ◽  
Vol 28 (04) ◽  
pp. 673-695 ◽  
Author(s):  
Gareth Tracey ◽  
Gunnar Traustason
Keyword(s):  
Nilpotent Radical ◽  
Engel Elements ◽  
Locally Nilpotent ◽  
Engel Element

Let [Formula: see text] be a group and let [Formula: see text] be a left [Formula: see text]-Engel element of order dividing [Formula: see text]. Suppose furthermore that [Formula: see text] has no elements of order [Formula: see text], [Formula: see text] and [Formula: see text]. We show that [Formula: see text] is then contained in the locally nilpotent radical of [Formula: see text]. In particular, all the left [Formula: see text]-Engel elements of a group of exponent [Formula: see text] are contained in the locally nilpotent radical.

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