An example of a Q-minimal precompact topological group containing a nonminimal closed normal subgroup

1979 ◽  
Vol 27 (3) ◽  
pp. 323-327 ◽  
Author(s):  
Ulrich Schwanengel
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Closed Normal Subgroup

Classification of Demushkin Groups

1967 ◽  
Vol 19 ◽  
pp. 106-132 ◽  
Author(s):  
John P. Labute
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Bilinear Form ◽  
Minimal System ◽  
List Type ◽  
Type Number ◽  
Cup Product ◽  
Minimal System Of Generators ◽  
Closed Normal Subgroup

A pro-p-group G is said to be a Demushkin group if(1)dimFp H1(G, Z/pZ) < ∞,(2)dimFp H2(G, Z/pZ) = 1,(3)the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro-p-group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F9 (F, F); cf. §1.4.

scholarly journals Groups – Additive Notation

2015 ◽  
Vol 23 (2) ◽  
pp. 127-160 ◽  
Author(s):  
Roland Coghetto
Keyword(s):  
Topological Group ◽  
Group Theory ◽  
Abelian Group ◽  
Normal Subgroup ◽  
Dense Subset ◽  
Abelian Topological Group ◽  
Covering Group ◽  
Mizar Mathematical Library ◽  
Order Of An Element ◽  
Generic Theory

Abstract We translate the articles covering group theory already available in the Mizar Mathematical Library from multiplicative into additive notation. We adapt the works of Wojciech A. Trybulec [41, 42, 43] and Artur Korniłowicz [25]. In particular, these authors have defined the notions of group, abelian group, power of an element of a group, order of a group and order of an element, subgroup, coset of a subgroup, index of a subgroup, conjugation, normal subgroup, topological group, dense subset and basis of a topological group. Lagrange’s theorem and some other theorems concerning these notions [9, 24, 22] are presented. Note that “The term ℤ-module is simply another name for an additive abelian group” [27]. We take an approach different than that used by Futa et al. [21] to use in a future article the results obtained by Artur Korniłowicz [25]. Indeed, Hölzl et al. showed that it was possible to build “a generic theory of limits based on filters” in Isabelle/HOL [23, 10]. Our goal is to define the convergence of a sequence and the convergence of a series in an abelian topological group [11] using the notion of filters.

scholarly journals $FC^-$-elements in totally disconnected groups and automorphisms of infinite graphs

2003 ◽  
Vol 92 (2) ◽  
pp. 261 ◽  
Author(s):  
Rögnvaldur G. Möller
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Automorphism Groups ◽  
Locally Compact Group ◽  
Infinite Graphs ◽  
Theoretical Interpretation ◽  
Locally Compact ◽  
Compact Closure ◽  
Totally Disconnected ◽  
Compactly Generated

An element in a topological group is called an $\mathrm{FC}^-$-element if its conjugacy class has compact closure. The $\mathrm{FC}^-$-elements form a normal subgroup. In this note it is shown that in a compactly generated totally disconnected locally compact group this normal subgroup is closed. This result answers a question of Ghahramani, Runde and Willis. The proof uses a result of Trofimov about automorphism groups of graphs and a graph theoretical interpretation of the condition that the group is compactly generated.

On theorems of Thornton

1972 ◽  
Vol 6 (2) ◽  
pp. 211-212 ◽  
Author(s):  
R. Lalithambal
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Discrete Group ◽  
Torsion Group ◽  
Additional Hypothesis

The topology of a topological group in which the intersection of open sets is open is uniquely determined by a normal subgroup, and the group is uniquely an extension of an indiscrete group by a discrete group. This was proved by M.C. Thornton under the additional hypothesis that the group is a torsion group. The proofs here given make the more general facts almost trivial.

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.

scholarly journals Semi-free subgroups of a profinite surface group

2018 ◽  
Vol 21 (5) ◽  
pp. 901-910
Author(s):  
Matan Ginzburg ◽  
Mark Shusterman
Keyword(s):  
Normal Subgroup ◽  
Surface Group ◽  
Infinite Index ◽  
Closed Normal Subgroup ◽  
Free Subgroups

Abstract We show that every closed normal subgroup of infinite index in a profinite surface group Γ is contained in a semi-free profinite normal subgroup of Γ. This answers a question of Bary-Soroker, Stevenson, and Zalesskii

scholarly journals Recapturing semigroup compactifications of a group from those of its closed normal subgroups

2000 ◽  
Vol 23 (1) ◽  
pp. 37-43
Author(s):  
M. R. Miri ◽  
M. A. Pourabdollah
Keyword(s):  
Normal Subgroup ◽  
Compact Group ◽  
Locally Compact Group ◽  
Normal Subgroups ◽  
Semitopological Semigroup ◽  
Locally Compact ◽  
Semigroup Compactifications ◽  
Closed Normal Subgroup

We know that ifSis a subsemigroup of a semitopological semigroupT, and𝔉stands for one of the spaces𝒜𝒫,𝒲𝒜𝒫,𝒮𝒜𝒫,𝒟orℒ𝒞, and(ϵ,T𝔉)denotes the canonical𝔉-compactification ofT, whereThas the property that𝔉(S)=𝔉(T)|s, then(ϵ|s,ϵ(S)¯)is an𝔉-compactification ofS. In this paper, we try to show the converse of this problem whenTis a locally compact group andSis a closed normal subgroup ofT. In this way we construct various semigroup compactifications ofTfrom the same type compactifications ofS.

Groups with Equal Uniformities

1969 ◽  
Vol 21 ◽  
pp. 655-659 ◽  
Author(s):  
R. T. Ramsay
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Vector Group ◽  
Locally Compact ◽  
Left And Right ◽  
The Relationship

If G = (G, τ) is a topological group with topology τ, then there is a smallest topology τ* ⊇ τ such that G* = (G, τ*) is a topological group with equal left and right uniformities (1). Bagley and Wu introduced this topology in (1), and studied the relationship between Gand G*. In this paper we prove some additional results concerning G* and groups with equal uniformities in general. The structure of locally compact groups with equal uniformities has been studied extensively. If G is a locally compact connected group, then G has equal uniformities if and only if G ≅ V× K,where F is a vector group and Kis a compact group (5). More generally, every locally compact group with equal uniformities has an open normal subgroup of the form V× K(4).

scholarly journals Galois groups as quotients of Polish groups

2020 ◽  
Vol 20 (03) ◽  
pp. 2050018
Author(s):  
Krzysztof Krupiński ◽  
Tomasz Rzepecki
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Galois Group ◽  
Galois Groups ◽  
Topological Dynamics ◽  
Strong Type ◽  
Polish Groups ◽  
Complete Type ◽  
Definable Groups ◽  
Countable Theory

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an [Formula: see text] normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over [Formula: see text]. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math. 228 (2018) 863–932] which says that for any strong type defined on a single complete type over [Formula: see text], smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the paper mentioned above about bounded quotients of type-definable subgroups of definable groups.

scholarly journals Inducing Super-Approximation

2019 ◽  
Author(s):  
Alireza Salehi Golsefidy ◽  
Xin Zhang
Keyword(s):  
Normal Subgroup ◽  
Positive Integer ◽  
Spectral Gap ◽  
Zariski Closure ◽  
Connected Component ◽  
Finitely Generated ◽  
Closed Normal Subgroup

Abstract Let $\Gamma _2\subseteq \Gamma _1$ be finitely generated subgroups of ${\operatorname{GL}}_{n_0}({\mathbb{Z}}[1/q_0])$ where $q_0$ is a positive integer. For $i=1$ or $2$, let ${\mathbb{G}}_i$ be the Zariski-closure of $\Gamma _i$ in $({\operatorname{GL}}_{n_0})_{{\mathbb{Q}}}$, ${\mathbb{G}}_i^{\circ }$ be the Zariski-connected component of ${\mathbb{G}}_i$, and let $G_i$ be the closure of $\Gamma _i$ in $\prod _{p\nmid q_0}{\operatorname{GL}}_{n_0}({\mathbb{Z}}_p)$. In this article we prove that if ${\mathbb{G}}_1^{\circ }$ is the smallest closed normal subgroup of ${\mathbb{G}}_1^{\circ }$ that contains ${\mathbb{G}}_2^{\circ }$ and $\Gamma _2\curvearrowright G_2$ has spectral gap, then $\Gamma _1\curvearrowright G_1$ has spectral gap.

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