Normal subgroups of locally compact groups

1976 ◽  
Vol 16 (5) ◽  
pp. 767-771
Author(s):  
M. I. Kabenyuk
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact

scholarly journals LOCALLY NORMAL SUBGROUPS OF TOTALLY DISCONNECTED GROUPS. PART II: COMPACTLY GENERATED SIMPLE GROUPS

2017 ◽  
Vol 5 ◽  
Author(s):  
PIERRE-EMMANUEL CAPRACE ◽  
COLIN D. REID ◽  
GEORGE A. WILLIS
Keyword(s):  
Boolean Algebras ◽  
Simple Groups ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact ◽  
Composition Factors ◽  
Totally Disconnected ◽  
Compactly Generated ◽  
Centralizer Lattice

We use the structure lattice, introduced in Part I, to undertake a systematic study of the class $\mathscr{S}$ consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are nondiscrete. Given $G\in \mathscr{S}$, we show that compact open subgroups of $G$ involve finitely many isomorphism types of composition factors, and do not have any soluble normal subgroup other than the trivial one. By results of Part I, this implies that the centralizer lattice and local decomposition lattice of $G$ are Boolean algebras. We show that the $G$-action on the Stone space of those Boolean algebras is minimal, strongly proximal, and microsupported. Building upon those results, we obtain partial answers to the following key problems: Are all groups in $\mathscr{S}$ abstractly simple? Can a group in $\mathscr{S}$ be amenable? Can a group in $\mathscr{S}$ be such that the contraction groups of all of its elements are trivial?

scholarly journals Totally disconnected, nilpotent, locally compact groups

1997 ◽  
Vol 55 (1) ◽  
pp. 143-146 ◽  
Author(s):  
G. Willis
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact ◽  
Totally Disconnected ◽  
Compactly Generated

It is shown that, if G is a totally disconnected, compactly generated and nilpotent locally compact group, then it has a base of neighbourhoods of the identity consisting of compact, open, normal subgroups. An example is given showing that the hypothesis that G be compactly generated is necessary.

A New Group Algebra for Locally Compact Groups II

1965 ◽  
Vol 17 ◽  
pp. 604-615 ◽  
Author(s):  
John Ernest
Keyword(s):  
Group Algebra ◽  
Von Neumann Algebra ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact ◽  
Central Projections ◽  
Von Neumann ◽  
One To One

In an earlier work, we defined and described a new group algebra , which is a von Neumann algebra containing the group G (3). In this paper we continue this study be relating the lattice of normal subgroups of the group G to the lattice of central projections of the group algebra . More precisely, we shall exhibit a one-to-one mapping ϕ of the lattice of closed normal subgroups of G into the lattice of central projections of , having the property that if N1 ⊂ N2, then ϕ(N2) ≤ ϕ(N1).

scholarly journals Locally normal subgroups of simple locally compact groups

2013 ◽  
Vol 351 (17-18) ◽  
pp. 657-661 ◽  
Author(s):  
Pierre-Emmanuel Caprace ◽  
Colin D. Reid ◽  
George A. Willis
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact

scholarly journals Amenable uniformly recurrent subgroups and lattice embeddings

2020 ◽  
pp. 1-38
Author(s):  
ADRIEN LE BOUDEC
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact

We study lattice embeddings for the class of countable groups $\unicode[STIX]{x1D6E4}$ defined by the property that the largest amenable uniformly recurrent subgroup ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is continuous. When ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ comes from an extremely proximal action and the envelope of ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is coamenable in  $\unicode[STIX]{x1D6E4}$ , we obtain restrictions on the locally compact groups $G$ that contain a copy of $\unicode[STIX]{x1D6E4}$ as a lattice, notably regarding normal subgroups of  $G$ , product decompositions of  $G$ , and more generally dense mappings from $G$ to a product of locally compact groups.

scholarly journals A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Colin D. Reid
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Normal Subgroups ◽  
Locally Compact ◽  
Derived Length ◽  
Chief Factors ◽  
Compactly Generated

AbstractWe classify the locally compact second-countable (l.c.s.c.) groups 𝐴 that are abelian and topologically characteristically simple. All such groups 𝐴 occur as the monolith of some soluble l.c.s.c. group 𝐺 of derived length at most 3; with known exceptions (specifically, when 𝐴 is \mathbb{Q}^{n} or its dual for some n\in\mathbb{N}), we can take 𝐺 to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups.

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