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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Particular property of an open subsemigroup in a given topological group

In chapter $\rm{III}$ of the "Topologie générale" of Bourbaki -- which is the chapter they dedicate to a brief introduction to generalities concerning topological groups -- I have ...
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Concept of a topological group [closed]

I have read the definition multiple times and have skimmed a few posts here, yet I still cannot fully grasp the idea of a topological group. Would anyone care to explain it with a few examples?
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understanding a step in proof of strong approximation for $SL_2$

Let $k$ be a global field and $k_v$ denote its completion at a place $v$. Let $Z$ be the closure of $SL_2(k)$ in $SL_2(\mathbb{A})$; $Z$ is a subgroup. Assume $Z$ contains $SL_2(\mathcal{O}_v)$ for ...
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$H$ closed implies $G / H$ Hausdorff

Let $G$ be a topological group and $H$ be a subgroup of $G$. It is to show that the closedness of $H$ implies $G/H$ being a Hausdorff space. We denote the projection map $G \to G/H$ with $p$. My book ...
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Must any $\phi \in \operatorname{Hom}_G(V, L^2(G))$ have continuous values?

Let $G$ be a compact group and $V$ a finite-dimensional vector space with a continuous $G$-action. Consider a linear map $\phi: V \to L^2(G)$ satisfying that for any $v \in V, h \in G$: $$ \phi(v)(...
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must any $\phi \in \operatorname{Hom}_G(V, L^2(G))$ be continuous?

Let $G$ be a compact group and $V$ a finite-dimensional vector space with a continuous $G$-action. Consider a linear map $\phi: V \to L^2(G)$ satisfying that for any $v \in V, h \in G$: $$ \phi(v)(...
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Is there a group defined over a set of reals whose elements change the real numbers while preserving the sum of the reals in the set?

Is there a continuous, (maybe smooth?) group that is defined over a finite set of zeros and/or positive real numbers (say the set of $3$ real numbers $\{1.59, 105, 19.42\}$) whose elements are binary ...
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Sharply $k$-transitive actions on spheres

A nice fact from complex analysis is that the mobius group acts sharply 3-transitively on the Riemann sphere. I am wondering if other sharply k-transitive (continuous) actions are known on any $S^n$, ...
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Co-compact lattice in locally compact hausdorff groups

Bekka-Mayer in their book below, in II.2 says: If $\Gamma$ is a discrete and cocompact subgroup of a locally compact group $G$, then $\Gamma$ is a lattice in $G$. I can't seem to prove it. Of course ...
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$f,g: X \to A$ be continuous functions then $f+g$ is also continuous.

Let $X$ be a topological space and $A$ be an abelian group with discrete topology. If $f,g: X \to A$ are continuous functions then how do I show that $f+g$ is also continuous ? So I need to show that $...
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Show that a metrizable Abelian topological group can be metrized by an invariant metric $d$.

My efforts: Let the Abelian topological group $G$ be metrized by $\rho$, which is not invariant. We want to construct an invariant metric $d$ from $\rho$. Define $\widetilde{\rho}(x,y):=$ max{$\rho(x,...
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How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?

Preliminaries An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}(X)$ such that: $\emptyset\in\mathcal{X}$. For $A,B\in\mathcal{X}$ then $A\cup B\in\mathcal{X}$. For $A,B\in\...
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Proof that the set $\{ m + n \sqrt{2} : m,n \in \mathbb{Z} \}$ is dense in $\mathbb{R}$

I think I have a correct proof of the above statement. If it is wrong, I'd like to know why. If it is correct, any suggestions on how to make the proof more concise would be greatly appreciated. Here ...
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Connected components of a coset space

Let $G$ be a topological group and $H\leq G$ be a subgroup. Let $p\;\colon G\to\pi_0\left(G\right)$ be the quotient map, $G_0$ be the identity path component of $G$ and $\left(G/H\right)_0$ be the ...
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A question about Hausdorff quotient by Group Action. Also, there are some very related material in Lee's book.

Let $G$ be a group acting on a topological space $X.$ I wanted to check the following statement. "Recall that the action is called even (or classically properly discontinuous) if for every point ...
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Showing that the additive group of a local field is locally profinite

Let $F$ be a non-archimedean local field, $\mathcal{O}$ its integers, $\mathfrak{p}$ its maximal ideal and $k := \mathcal{O}/\mathfrak{p}$ its residue field with $|k| = q$. Let $\pi$ be a uniforimser ...
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Product of a closed subgroup and a lattice in a Lie group

Let $H$ be a closed connected subgroup of the Lie group $G$ and $\Gamma$ be a lattice in $G$. Furthermore, assume that $H\cap \Gamma$ is a lattice in $H$. I wonder if it is true that the product $H\...
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What is a topological group on $ \mathbb{R}^n\ $

While working on another problem, I have to use the idea of the topological group on $ \mathbb{R}^n\ $. The on-line definitions don’t help much because they just say something like “$ \mathbb{R}^n\ ...
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CW structure on $SO(n)$

Surprisingly I didn’t find such a question. I want to construct a cellular (CW) structure on SO(n) groups. I can do it for $n = 2$ and $n= 3$ because those are homeomorphic to the circle and ...
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1answer
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Closed subgroup and closed orbit

Let $G$ be a topological group and $\Gamma$ be a lattice in $G$. Consider the action of $G$ on the homogeneous space $G/\Gamma$ by left translation. Let $F$ be a closed subgroup of $G$ and $x\in G/\...
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Subsets of positive measure in a group

Let $G$ be a Lie group (maybe LCH and second countable topological group is enough here) and $\mu$ be a Borel probability measure on $G$ (not necessarily Haar). Suppose $U$ is an open subset of $G$ of ...
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Is the group action by a topological homeomorphism group continuous?

Let $F$ be a topological space such that $G := Homeo(F)$ is a topological group in the compact-open topology. Let $\phi : G \times F \to F$ sending $(g,x)$ to $g(x)$ be the map for the action of $G$ ...
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Support of the convolution of two measures

Let $\mu, \nu$ be two Borel Probability measures on a Lie group $G$. I wonder if it is true that $$supp(\mu \ast \nu)=supp(\mu) \cdot supp(\nu),$$ or at least we have one contained in another? Recall ...
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Uniform parametrisation of $SO(3)$

Does there exist a map from a hypercube of some dimension to $SO(3)$ such that pushforward measure of standard measure on the hypercube is invariant under action of $SO(3)$ on itself?
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A topological detail in the definition of Lie group

A Lie group $ G $ is an $r$-times-differentiable manifold endowed with a group structure, i.e. with an associative binary operation $$ \mu:\quad G\times G \longrightarrow G :\qquad\left\{x\,, y\right\...
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Countable basis of compact open subgroups in locally profinite group.

I am currently reading through a proof of the existence of a right haar measure for locally profinite groups. We have a locally profinite group $G$ with the assumption that for all compact open ...
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Orientation of a basis and of a vector

Let $\mathbb{R}^n$ be the Euclidean space of dimension $n.$ We say that two ordered bases have the same orientation whenever the change-of-base matrix has determinant $>0.$ This determines an ...
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1answer
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Haar measure on $\operatorname{GL}_n(\mathbb{R})_{+}$

Let's consider the group $$\operatorname{GL}_n(\mathbb{R})_{+} = \left\{ M \in M_n(\mathbb{R}) \mid \det(M) > 0\right\}$$ We identify this set as a open subset of $\mathbb{R}^{n^2}$. It is known ...
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Alternative definition of inverse limit implies the universal property

When I read about Inverse Systems and Inverse Limits (specifically about Profinite Groups) for the first time, I learn the following definition: Definition 1. An inverse limit $(G,f_i)$ of an inverse ...
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1answer
107 views

Show $x \mapsto \int_G f(yx) \mu(dy)$ is continuous

This is exercise 11.2 in Folland's book: Let $\mu$ be a Radon measure on the locally compact group $G$ and $f \in C_c(G)$. Prove that $$x \mapsto \int_G f(yx) \mu(dy)$$ is continous. Before ...
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Noncompactness of $SL_n(\mathbb R)/SL_n(\mathbb Z)$

For $n\ge 2$, I saw from a paper that $SL_n(\mathbb R)/SL_n(\mathbb Z)$ is noncompact. I wonder how to prove this fact. If the proof for the general case $n\ge 2$ is too complicated, I am also glad to ...
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Compact abelian group is profinite if and only if it almost has finite exponent

I am looking for a reference for the following fact: If $G$ is a compact Hausdorff abelian group, then it is profinite if and only if for each open $V\ni 0$, there is some $n>0$ such that $V\...
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About definition and isomorphism of completion. (Topological group)

I studied a completion of topological group by textbook Atiyah. : What I just understand is if $G$ is a topological group, then if $\hat{G}$ is a set of equivalent Cauchy sequence, then we call $\hat{...
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Prove that the composition map is continuous with respect to the metric topology on $\operatorname{Iso}(M)$

Let $M$ be a finite dimensional Riemannian manifold and $\operatorname{Iso}(M)$ be its set of isometries. It can be shown that $\operatorname{Iso}(M)$ is a finite dimensional manifold with a metric as ...
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Continuity of operations in topological groups

I have a question regarding continuity in topological groups. From Wikipedia: A topological group $G$ is a topological space that is also a group such that the group operation: \begin{align*} \mu: G \...
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1answer
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Differentiable homomorphism $f: U \rightarrow V$ has constant rank

Let $U\subseteq \mathbb R^m$ open and suppose that a group structure is defined on $U$ with multiplication $\mu: U\times U \rightarrow U$ of class $C^k,k\geq 1$. Suppose that $V\subseteq \mathbb R^n$ ...
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A connected group which is disconnected

It is known that $G = \mathrm{SL}(2, \mathbb{R})$ is a connected group. Consequently $G$ cannot have any open subgroups. (Any such subgroup would be closed as well, contradicting connectedness.) How ...
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A fallacious argument concerning certain topological groups

Let $K$ be a non-Archimedean local field, and let $\mathcal{O}$ be its ring of integers. Let $U$ be the group of $4 \times 4$ unipotent upper-triangular matrices over $K$. Consider the topology on $K$ ...
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Approximating convolution with $L^1$ function by sum of translation operators

Let $G$ be a locally compact topological group. Let $\mu$ be left Haar measure on G. Suppose $f \in L^1(G)$. Then convolution with $f$ defines an operator from left uniformly continuous and bounded ...
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Haar measure of infinite many products.

Consider compact groups $G_i$, not necessarily countable, and denote $\mu_i$ by their nonzero unique Haar measures. By the theorem of Tychonoff, the product space $\Pi G_i$ is compact. So $\Pi G_i$ is ...
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Weil's completion of locally totally bounded groups to locally compact groups

Let $G$ be a topological group. Say that $A\subseteq G$ is totally bounded if for any neighborhood of $e$, $U$, there exists a finite subset $F\subseteq G$ such that $A\subseteq FU$. $G$ is locally ...
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Is there a deep reason why $\mathbb{C}$ is a topological field w.r.t. the Euclidean norm?

Today I realised that I didn’t remember how to prove that the complex numbers are a topological field w.r.t. to the metric induced by the Euclidean norm $\|(x,y)\|=\sqrt{x^2+y^2}$. I think the usual ...
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Topological groups vs regular groups [duplicate]

I know group theory and I'm familiar with the concept and definition of Group. Today I was reading an article about topology and discoverer the concept of "topological group". I read the ...
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Kernel of restriction to dense subgroup is again dense

Let $X$ and $Y$ be topological compact abelian groups, and let $A$ and $B$ be dense subgroups of $X$ and $Y$ respectively. Let $\varphi\colon X \to Y$ be a continuous homomorphism such that $\varphi(A)...
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Intersections of open cosets in a profinite group

Let $G$ and $H$ be profinite groups and $\varphi\colon G \to H$ be a continuous homomorphism. It is true that $$\ker(\varphi) = \bigcap_{V \leq_o G} V$$ where $V$ runs trough the set of open subgroups ...
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When is convolution not commutative?

Let $G$ be a locally compact Hausdorff group with a left Haar measure $\lambda$. Define the convolution of two functions $f,g \in L^1(G)$ by $$(f \ast g)(x) = \int f(y) g(y^{-1}x) d\lambda (y), ~~~ \...
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True or false: infinite sequence in a compact topological group is dense. [duplicate]

This is from an exercise in Bredon's Topology and Geometry: Let $G$ be a compact topological group (assumed to be Hausdorff). Let $g\in G$ and define $A=\{g^n:n=0,1,2...\}$. Then show that the ...
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1answer
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Writing G as a product of groups

$G$ is a connected, locally compact group satisfying the second axiom of countability, and $C$ is a discrete central subgroup of $G$ such that $G/C$ is compact. In the book I am reading (Varadarajan, ...
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Verify that $f_{n} \rightarrow f_0$, where $f_{n}=0 \ \forall \ x \in[0,1]$ [closed]

Let $C[0,1]$ and $d$ be defined on this set by $$d(f, g)=\int_{0}^{1}|f(x)-g(x)| \, \mathrm d x$$ where $f$ and $g$ are in $C[0,1]$. Define a sequence of functions$f_{1}, f_2, \ldots, f_{n} \ldots$ in ...
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1answer
60 views

Ordered Groups: Left Multiplication vs Right Multiplication

Given that $G$ is a linearly ordered group (bi-ordered). I want to try and understand the difference between the “size” of left multiplication vs right multiplication (which I have written below using ...

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