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Questions tagged [topological-groups]

A topological group is a group endowed with a topology such that the group operation and inversion are continuous maps. They are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis.

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Intersection of connected components with discrete subgroup

I am currently studying Harmonic Analysis and didn't quite understand part of the proofs for the structure theorems of locally-compact abelia groups (LCA). Let $A$ be an LCA group such that $\hat{A}/\...
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The circle S is a topological group, as is the torus T=S*S={(cos2πθ+isin2πθ), cos2πφ+isin2πφ) | 0≤θ, φ≤1} [duplicate]

The circle S is a topological group, as is the torus T=S*S={(cos2πθ+isin2πθ), cos2πφ+isin2πφ) | 0≤θ, φ≤1}. Let H be the subgroup defined by φ=αθ, where α is an irrational number. Show that H is not ...
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CW Structure on Self-Homeomorphism Groups

If $X$ is a finite CW complex then according to Milnor's article On Spaces Having the Homotopy Type of a CW-Complex, the mapping space $$Map(X,X)$$ (which we furnish with the compact-open topology) ...
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Is the usual metric on $\Bbb{N}^\Bbb{N}$ left invariant on $S(\Bbb{N})$?

Let $\Bbb{N}^\Bbb{N}$ be the set of all functions $(x_n\mid n\in\Bbb{N})$ from $\Bbb{N}$ into itself (I identify sequences with their images, as usual). I know this is a metrizable space with ...
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Suggestion for book on topological group and other topics.

What is a great book where I can find the following topics: 1. Topological groups; 2. Topological properties of orthogonal group $O(n)$; 3. Topological properties of special orthogonal ...
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Every quasi-invariant measures is in an invariant measure class

I'm reading "Ergodic Theory and Semisimple Groups" by Zimmer and at the very beginning of Chapter $2$ (pp. $8$) the author claims that An action with quasi-invariant measure can be thought of as ...
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Continuous group action on a finitely generated group and compact Hausdorff space

Let $\varphi:G\times X\to X$ be a continuous action such that $G$ be a finitely generated group, $X$ be a compact Hausdorff space, and $\mathcal{U}=\{U_i\}_{i=1}^m$ be a finite open cover of $X$. ...
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How is the base of the topology on a profinite group?

If $G$ and $H$ are profinite groups and $\varphi: G \to H$ is a homomorphism, then $\varphi$ is continuous iff the inverse image of any open normal subgroup in $H$ is open in $G$. If $\varphi$ is ...
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60 views

Automorphisms group of a structure is a closed subgroup of the permutations over $\mathbb{N}$

Here is the second part of Example $7)$ of Kechris' "Classical Descriptive Set Theory" (pp. $59-60$): More generally, consider a structure $$\mathcal{A}=(A,(R_i)_{i\in I},(f_j)_{j\in J},(c_k)_{k\...
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Ideal in Integral Adelic Space?

Suppose that $\boldsymbol{x}_i,\boldsymbol{y}_i$, $1\le i\le m$, lie in integral adelic $m$-space $\mathbb{A}_\mathbb{Z}^m = (\widehat{\mathbb{Z}}\times\mathbb{R})^m$ and that $\bigoplus\limits_{i=1}^...
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Topology of a free group

I wonder is there any general properties/ restrictions to the possible topologies of a free group (to make it a topological group ofc). More generally do such restrictions exist for any group written ...
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Absolute convergence of Fourier series of periodic adelic function

I've asked several questions about this topic before, including here and here. I've gotten many helpful responses, but I still do not completely understand what is going on. Let $f: \mathbb A_{\...
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1answer
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Cohomological dimension of a topological group with torsion

I'm interested in a proof (or counter-example) of the following: Let $G$ be a topological group. If $G$ contains torsion then $H^n(BG)\neq 0$ for infinitely many $n$. I know this is true for ...
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Number System with Torsion

Introduction: A number system, long known but seldom seen, is (re)introduced for which some elements are torsion and some are torsion-free. A topology question and an analytic number theory question ...
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Uniform continuity of $f(x):=\inf\{r \in D : x \in U_r\}$

I have a sequence of sets $\{ U_r\}_{r\in D}$ in a topological group $G$, where $D$ is the set of dyadic numers in $[0,1]$ such that $U_{1/2^n}$ is a neighborhood of the identity $e \in G$, $U_{1/2^n}...
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On the algebraic structures of $C_p(X,Y)$

Recently I started studying the connections between the algebraic structures and functional spaces. I'm interested in the space of all continuous functions $C_p(X,Y)$ endowed with pointwise topology....
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Properties of a Hausdorff topological group (“proof” verification).

Problem. The centralizer $C_{G}(S)$ of a subset $S$ of a group $G$ is defined by $$C_{G}(S) = \{g \in G \mid sg = gs\mathrm{\,for\,all\,}s \in S\}$$ and the normalizer $N_{G}(H)$ of a subgroup $H$ ...
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1answer
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Use the theory of characters to derive the following relation for the representations of $SU_{2}.$

The question is given below: And the hint at the back of the book says: Establish the corresponding equality for characters. And this was a question I was helped on it, which establish the relation ...
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Winding number in 4D & SU(2) group

In the book 'Quantum field theory' by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such ...
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Prüfer Groups and Product Topologies

For each $p\in\mathbb{P}$, the Prüfer group $\mathbb{Z}(p^{\infty})$ is a divisible abelian group which can be given the discrete topology or the topology it inherits via its identification as a ...
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If quotient by a closed normal subgroup is discrete

Suppose that $T$ is a topological group and $K$ is a closed normal subgroup. Is $T/K$ a discrete space? I think it is, since $tK$ for all $t\in T$ is closed, i.e. every singleton $\{tK\}$ is closed. ...
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Representation of $\overline{\mathbb{Q}}$ in One Dimension

Introduction: Below gives an approach to realize the algebraic numbers $\overline{\mathbb{Q}}\subseteq\mathbb{C}$ within a $1$-dimensional compact connected abelian group (solenoid). Essentially a ...
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Why is the stabilizer of a line segment a cylinder?

There is a passage in a paper I'm reading discussing the stabilizer of an edge. For an edge (passing through the origin), its stabilizer (in $\operatorname{GL}_2(\mathbb{R})$) must fix the ...
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Lattice and abelian Lie groups

Let $\Lambda$ be a discrete (lattice) subgroup of $\mathbb R^n$. Let $V:=\langle \Lambda\rangle_\mathbb R$. Define the abelian Lie group $G:=V/\Lambda$. Now if $H$ is a Lie subgroup of $G$. Does ...
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1answer
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If a set is a dense or relatively dense subset of a topological group

Suppose that $X$ is a topological space, and $T$ is a topological group which continuously acts on $X$ on the right. We call the pair $(X,T)$ a (right) transformation group. We know that $(X,\mathbb ...
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Topology “generated by” normal subgroups (Topological groups).

Let $G$ be a group and $L$ be a non-empty family of normal subgroups such that if $K_{1},K_{2} \in L$ and $K_{3}$ is a normal subgroup containing $K_{1} \cap K_{2}$ then $K_{3} \in L$. Let $T$ be a ...
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Representation of locally compact groups

It is well-known that every compact Hausdorff group is the inverse limit of compact Lie groups (in fact, compact linear groups). Question: is there any analogue for locally compact (or at least lcsc) ...
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1answer
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finiteness of an abelian topological group

Let A be an abelian (Hausdorff) topological group. Assume that (1) the set of its torsion elements, and (2) a finitely generated subgroup are dense subsets of A. My question: must A be finite? (...
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A couple of questions about admissible representations and $K$-finite vectors

Let $G$ be the points of a connected, reductive group over $\mathbb R$, and let $K$ be a maximal compact subgroup of $G$. Let $(\pi,V)$ be a continuous representation of $G$ on a Hilbert space for ...
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Maximal compact subgroup of a nilpotent Lie group

Let $G$ be a connected nilpotent Lie group and $K \subset G$ a maximal compact subgroup. Can we prove that $K$ is always normal?
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Group Algebra of a Discrete Group and Different Notions of a Group Algebra?

I am reading these two wiki articles: https://en.wikipedia.org/wiki/Group_algebra and https://en.wikipedia.org/wiki/Pontryagin_duality From my understanding, the group algebra of a topological group ...
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1answer
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A homomorphism which is trivial on the $n$-torsion must have an $n$'th root?

Suppose that $(G,+)$ is a locally compact abelian group. Let $G_n:=\{g\in G : ng=0_G\}$ and let $\chi:G\rightarrow S^1$ be a continuous character. Assuming that $\chi(G_n) = 1_{S^1}$. Is it ...
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$S_\infty$ is a non-locally compact Polish group (Kechris)

Here is example $7)$, pp. $59$ of Kechris' book "Classical Descriptive Set Theory": Let $S_\infty$ be the group of permutations of $\mathbb{N}$. With the relative topology as a subset of $\...
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Question 4, chapter III, section 7 in Vinberg “Linear representations of groups. ” [closed]

The question and its answer is given below: Where T is the unit circle, and $\Phi_{n}$ is described below: But I do not understand the solution,could anyone explain it for me please? or give me ...
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1answer
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prove that any central function of $SU_{2}$ is uniquely determined by its restriction to the following subgroup.

The question is given below: But I do not know how to solve it, could anyone give me a hint please? EDIT:
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Direct proof that closed subgroups of profinite groups are profinite

Each of the references that I check prove first the relatively involved characterization of profinite groups as compact Hausdorff totally disconnected topological groups, and then appeal to the fact ...
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1answer
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Compact normal subgroups of connected groups

We know that for a connected locally compact group $G$ there exists a compact normal subgroup $K$such that $G/K$ is a Lie group. Now, if $G$ is compact, could we find a proper compact normal subgroup $...
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Why are uniform spaces important?

Why are uniform spaces important? I've thought of two possible good answers: Topological groups. According to nCat locally compact groups are complete with respect to the left/right uniformity, and ...
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normal subgroup basis of $\mathrm{GL}(n, \mathbb{Q}_{p})$

Consider the group $G=\mathrm{GL}(n, \mathbb{Q}_p)$. This group is locally compact and totally disconnected, and we have a basis of open subgroups given by $$ K(p^m) = \left\{ \begin{pmatrix} a&b\...
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How to prove orthogonal group is compact with induction topology?

I would like to prove that $O(n,R)$ is a compact set. Can I just view $O(n,R)$ as a subset of $R^{n*n}$ and prove it is compact by proving it is bounded and closed?
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Continuity of representation of topological group

First, We set notations as follows. $G$ : topological group , $k$ : field , $V$ : linear topological space over $k$ , $\mathrm{Map}(V,V)$ : Set of all continuous maps from $V$ to $V$ $\mathrm{Aut}_k (...
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Left multiplication is homeomorphism of topological groups

This is a very simple question involving basic definitions. I want to prove that if $G$ is a topological group, left multiplication $f_a\colon g\mapsto ag$ is a homeomorphism of $G$. Clearly, this map ...
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G is homeomorphic to $G/H\times H$

Let G be a topological group and $H\le G$. Let $\pi: G\to G/H$ be the canonical projection and a continuous $\sigma: G/H\to G$ such that $\pi \circ \sigma = Id$. Prove that G is homeomorphic to $G/...
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$\ell^2$-convergence of convolution square root implies uniform convergence

I have problems understanding parts of a proof in [Proposition 18.3.5., Diximier, C*-algebras] for the special case of discrete groups. Let $G$ be a discrete group and let $\phi\in \ell^2(G)$. By 13....
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If $\chi(g)$ generates a dense subgroup of $\chi(G)$ for all $\chi$ then $g$ generates a dense subgroup of $G$.

This question arised from something in Ergodic theory, however this is not necessary to state or answer the question. Suppose that $G$ is a compact abelian group and $g\in G$. Are the following two ...
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Making the subgroup of units of a topological ring a topological group

I'm a beginner in the theory of topological algebra (I'm reading something about it in Robert's "A course in $p$- adic analysis"). At page 24, the author states that if $A$ is a topological ring, the ...
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1answer
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Unitary Representations and the Peter-Weyl Theorem

Consider part II of the Peter-Weyl Theorem (see this wikipedia page for more information): Let $\rho$ be a unitary representation of a compact $G$ on a complex Hilbert space $H$. Then $H$ splits ...
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1answer
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$\Bbb{Z}_px\subseteq\overline{\Bbb{Z}x}$

I'm studying $p$-adic integers and in the proof of the fact that closed subgroups of the additive group $\Bbb{Z}_p$ are ideals (see Robert's "A course in $p$-adic analysis", pp.23) I've found the ...
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35 views

Locally closed dense subgroup of a topological group coincides with the whole group

This question is very simple, but I don't get the right idea. Assume $H$ be a locally closed dense subgroup of a topological group $G$. Prove that $H=G$. I need to prove that $gH\cap H\...
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1answer
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If $H$ and $G/H$ are pseudocompact then is G also pseudocompact?

Theorem 5.25 in Edwin, Hewitt Abstract Harmonic Analysis Part 1, says if $H$ is a subgroup of a topological group $G$ and $H$ and $G/H$ are compact (resp. locally compact), then so is $G$. For my ...