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.

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.

Embeddings of locally compact abelian p-groups in Hawaiian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yanga Bavuma ◽  
Francesco G. Russo
Keyword(s):  
Algebraic Topology ◽  
Geometric Interpretation ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Hawaiian Earring ◽  
Path Connected

Abstract We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

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.

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.

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).

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