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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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group homomorphism from a profinite group continuous iff kernel open

I have a question regarding a (probably simple) fact. However I am lacking some basic topological knowledge. Let $G$ be a locally pro finite group, i.e. ever open neighborhood of $1_G$ contains a ...
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Locally compact nilpotent group has an open subgroup isomorphic to $\mathbb{R}^n\times K$

My question is about a possible generalization of the following structure theorem of locally compact abelian groups. Theorem: Let $G$ be a locally compact abelian group. Then here exists a compact ...
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Continuity of absolute value in topological ordered abelian groups

Let $(G,+,0)$ be an abelian topological ordered group, that is, $G$ is endowed with a total order $\leq$ such that, for any $a,b,c\in G$, we have that $a\leq b$ implies $a+c\leq b+c$. Moreover, $G$ is ...
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$(X/H)/(G/H)=X/G$? Here $G$ is a topological group acting continuously on $X$, $H$ is a closed normal subgroup of $G$.

Let $G$ be a topological group acting continuously on a topological space $X$ (on the left, denoted by $g\cdot x$, $g\in G, x\in X$), $H$ be a closed normal subgroup of $G$. Then The quotient ...
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Definition of the Fell topology: Completion with respect to a seminorm

I'm reading about the Fell topology and have a question on some preliminary material. My reference is these notes on automorphic representations. Let $G$ be a locally compact Hausdorff, second ...
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Covering of a topological group

Let $X$ be a topological group with a locally path-connected, path-connected covering $(\tilde{X},p)$. If we fix an $u\in p^{-1}(e)$, we should be able to deduce a unique group structure for $H=\tilde{...
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Is every topological groupoid equivalent to a disjoint union of topological groups?

It's a fact that any groupoid is equivalent to a disjoint union of (deloopings of) groups. See, e.g. Proposition 4.3 of https://ncatlab.org/nlab/show/groupoid#PropertiesEquivalencesOfGroupoids. Does ...
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Every point in completion of a topological group in closure of a countable subset

Is there a name or perhaps some interesting equivalent condition for the following condition on an abelian topological group $G$ with uniformity generated by the neighbourhoods of 0? Every point ...
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What is $SO(n+1)/O(n)$ as a topological space?

Consider $SO(n),O(n)$ as topological groups. Find out $SO(n+1)/O(n)$ as a topological space. My attempt: Observed the inclusion : $O(n) \hookrightarrow{} SO(n+1)$ by, $$A \mapsto\begin{bmatrix} det(...
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Prove that $[0,1]^n / S_n \cong \Delta^n$

Prove that $[0,1]^n / S_n \cong \Delta^n$, ( $S_n$ denotes the permutation group on n symbols, $\Delta^n:=\{(x_0,\dots,x_n)\in\Bbb R^{n+1} : x_0,\dots,x_n \ge 0 , x_0+\dots+x_n=1\}$, and the action is ...
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“Halving” neighborhoods in topological groups

Let $S=B\left(0,\frac{\epsilon}{2}\right)$ be a ball in $\mathbb{R}^n$.  Given a translate $S+t$, there is the nice property that, for all $x,y\in S+t$, we have $x-y\in S-S\subseteq B(0,\epsilon)$...
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Do dense topological subgroups always have the same finite quotients as their underlying bigger topological subgroup?

Definition: Let $G$ be a topological group. We call $G$ a finite quotient of $G$ if there exists a normal subgroup $N$ of $G$ such that $H = G/N$ and $H$ is finite. Let $G'$ be a dense subgroup of $G$...
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Hilbert's 5th problem and short exact sequences of Lie groups.

Let $G$ be a locally compact topological group. Suppose that there exists a normal closed subgroup $H$ which is also a Lie group such that $G/H$ with the quotient topology is also a Lie group. Prove ...
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Compactness of $PGL(n, \mathbb{R})$ and $PGL(n, \mathbb{C})$

I know that the $PGL(n, \mathbb{R})$ or $PGL(n, \mathbb{C})$ is Lie group, because $PGL(n, F) = GL(n, F) / Z(n, F)$, where $Z(n, F)$ - scalar transformation and $F$ is $\mathbb{C}$ or $\mathbb{R}$. ...
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Connected components of $PGL(n, \mathbb{R})$ ond $PGL(n, \mathbb{C})$

I know that the $PGL(n, \mathbb{R})$ or $PGL(n, \mathbb{C})$ is Lie group, because $PGL(n, F) = GL(n, F) / Z(n, F)$, where $Z(n, F)$ - scalar transformation and $F$ is $\mathbb{C}$ or $\mathbb{R}$. ...
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Unique characterizing property of identity component

Is there any characterizing property of path component of identity in a topological group? In other words, how will I show that some path connected subgroup is the identity path component of ...
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All connected closed subgroups in $\rm{SO}(3)$

I want to find all connected closed subgroups in $\rm{SO}(3)$. Is there a direct classification without attracting Lie algebras? Thank you very much!
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On proving that $G/\overline{\{e\}}$ is Hausdorff for a topological group $G$.

I'm working on the following problem, Let $(G,\tau)$ be an abelian topological group, and $H = \bigcap_{e \in U \in \tau}U$ be the intersection of open sets (or equivalently neighbourhoods) of the ...
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1answer
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Continuous group actions and transitivity

Let $G$ be a topological group and $M$ a topological space. Suppose that $$\circ: M\times G\longrightarrow M$$ is a continuous group action of $G$ on $M$. If $\circ$ is transitive, then we can take an ...
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The intersection of all stabilizers in a continuous action

Let $G$ be a topological group (locally compact and Hausdorff) continuously acting on a topological space $M$. Does this implies that the normal subgroup of $G$ made by the elements fixing each $m\in ...
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An example of a group with a topology

Do you know an example of a group with a topology satisfying both the following two conditions the product is separately continuous but not jointly continuous the inversion map is continuous.
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Exercise on categories of $G$-set

The following is exercise 6 from chapter I of Mac Lane and Moerdijk's Sheaves in Geometry and Logic: I'm currently having trouble with point $(b)$, probably because of my little ability with group ...
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The neighborhood of e in a topological group

suppse U is an open neighborhood of e in a topological group G and how to prove there exists an open neighborhood V s. t. $V=V^{-1}$ and $V^{2}$ is a subset of U? Another question is.how to prove the ...
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questions about connected and Hausdorff topological Space

1: let $X$ be a Hausdorff topological Space , $Y \subseteq X $ is Nonempty and dense in $X$. let $f: X \longrightarrow X $ be a continuous functions so that $ \forall y \in Y, f(y) = y $. Is $f$ a ...
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All Lie subgroups of $\mathbb{R}$

I want to describe all Lie subgroups of the real line. I know that it is the same as to describe all closed subgroups and I know how to do that for $S^1$. Next I wanted to consider $$\pi: \mathbb{R} \...
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A Corollary from Munkres' Topology

This exercise comes from Munkres' Topology, 2nd edition, Page 188. It says Corollary. Let $G$ be a topological group; let $A$ and $B$ be subsets of $G$. If $A$ is closed in $G$ and $B$ is compact, ...
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Is there an algebraic or geometric explanation for why $Spin_7\simeq_{\frac{1}{3}}S^7\times G_2$?

The octonions $\mathbb{O}$ are the $8$-dimension real (non-associative) normed division algebra. Forgetting the algebra structure leads to an identification of $\mathbb{O}$ with $\mathbb{R}^8$ as real ...
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If $G/H$ is homeomorphic to $\mathbb R^n$ then $H$ is normal subgroup

Let $G$ be a topological group and $H$ be a closed subgroup such that $G/H$ is homeomorphic (as a manifold) to the abelian group $\mathbb R^n$. Then, is it necessary that $H$ has to be normal ...
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Subgroups of Compact groups are exactly the totally bounded groups.

If $G$ is a compact topological group, then how can we prove that subgroups of $G$ are exactly the totally bounded groups. $\textbf{Note}$: We call a topological group $G$, totally bounded or ...
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If $G/H$ is compact what is $\pi_0(G/H)$

Let $G=R\times S$ be a connected, simply-connected linear subgroup of $GL(n,\mathbb R)$ such that $R$ is abelian central subgroup of $G$ and $S$ is a proper normal semisimple subgroup. Let $H$ be any ...
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Verification of morphism is $G-$module morphism.

Let $G$ be a topological group.(You can assume it is finite.) Let $A$ be a $G-$module which is also a topological abelian group equipped with $G-$action. Denote $\mathcal{C}^n(G,A)=\{f:G^{n}\to A\vert ...
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Discrete subgroup of topological group

I want to find textbook which contains the next proposition(?). I think it is true, but I can't find proof of that. Please teach me a textbook in which the next proposition(?) are proved. ...
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Non-Lie Subgroups of GLn(R)

Is there any subgroups of $GL_n(\mathbb{R})$ which are not Lie groups. Some trivial examples I know like $\mathbb{Q}^*\subset \mathbb{R}^*$ which are $0$-dimensional cases. I am looking for some ...
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Is there a LCA group $G$ such that $G/\mathbb{T}\cong\mathbb{R}$?

I'm looking for an example to the following situation: A locally compact abelian (LCA) group $G$ (I assume that the groups are Hausdorff) . A (closed) subgroup $H$ of $G$ which is isomorphic (as ...
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Intersection of orbits is the orbit of intersection

Let $G$ be a (topological) group acting on a space $X$. Let $x\in X$ and $A,B$ be two subgroups of $G$. Is it true in general that $(A\cdot x)\cap (B\cdot x)= (A\cap B)\cdot x$?
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Semisimple subgroups of $PSL_n(\mathbb C)$ are centerless

Since the protective special linear group $PSL_n(\mathbb C)$ is centerless. Is it true that any connected semisimple subgroup of $PSL_n(\mathbb C)$ is also centerless?
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Second isomorphism theorem for homogeneous spaces

So if a Lie group $G$ acts transitively on a manifold $X$, then $X$ is diffeomorphic to the quotient $G/H$ where $H$ is the stabilizer subgroup of a point $x$ in $X$. Now if $N$ is a normal ...
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Examples of Uniform Spaces

Are there any important examples of uniform spaces other than metric spaces and topological groups? Also, what is an example of when the uniform structure of topological groups is used?
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If $G$ is a LCA, does always exist a discrete subgroup $H$ of $G$ such that $G/H$ is compact?

Following an idea about proving Fourier inversion formula for arbitrary LCA groups from the corresponding result for compact groups, I stumbled upon this problem: is it true that, given an arbitrary ...
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The fundamental group of semisimple group orbits

It is known that the fundamental group of a compact semisimple Lie group is finite. Can this fact be generalized to compact orbits of semisimple Lie groups? i.e. Let $G$ be a semisimple Lie group such ...
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Is there a Borel-measurable projection to a closed subgroup

Suppose that $G$ is a compact metrizable group and let $H$ be a closed subgroup of $G$. Is it true that there must exists a Borel-measurable projection map $p:G\rightarrow H$ with the property that ...
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What is a complete topological group?

Given a metric space $ X $, one can forms its completion $ \hat{X} $. However, I have seen that one can also define completeness for topological groups. Can someone explain to me (as simply as ...
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Group structure on arbitrary topological spaces [duplicate]

Definition. Let $(G,\ast)$ be any group. Then $G$ will be said to be a topological group if there exists a topology on $G$ such that the map $f:G\times G\to G$ defined by $f(x,y)=xy^{-1}$ for all $(x,...
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Show that G is profinite

Let G be a compact group, H be an open subgroup of G. Show that if H is profinite, then G is also profinite. Lemma to use as a hint is this: Let G be a compact group and $ {N_i | i \in I}$ be ...
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Normal subgroups orbits

Let $G$ be a topological group acting transitively and effectively on the space $X$ and let $J,K$ be two normal subgroups of $G$ such that $G=J\cdot K$ and $J\cap K\not =\{e\}$. Let $Gx_0$ be the ...
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principal bundle of homogeneous spaces where the group is a product

Let $G$ be a connected topological group. Let $A,B$ be two closed normal subgroups such that $A$ is central. Suppose $A\cap B=\{e\}$ and $G=A\times B$ . Let $H$ be a closed subgroup. Consider the ...
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If $f \in \operatorname{c-Ind}_Q^P \sigma$, then $f|_N$ is compactly supported modulo $Q \cap N$?

From Casselman's notes on representation theory. If $p \in P$, it's not clear to me at all that $R_pf|_N$ must have compact support modulo $Q \cap N$. To prove this, we may assume $p = 1_P$. ...
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Quotient map $G \rightarrow H \backslash G$ is a principal fibre bundle: how to see the local trivializations?

Let $H$ be a closed subgroup of a Lie group $G$. Then $H$ acts on $G$ by left translation. The action is: (i): free (ii): proper Proof: (i) is clear. For (ii), we need to show that the preimage ...
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Closed Subgroup is Open in a certain Topological Group

In any topological group it is easy to show open subgroups are closed. Prove that the converse holds in the following situation: Let G be an abelian group, and let $$G=G_0 \supset G_1 \supset G_2\...
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Square integrable functions on Lie groups

If $\Gamma$ is a countably infinite discrete group and $F$ a finite subgroup of $\Gamma$, then one knows that there is an isomorphism of $F$-representations $l^2(\Gamma)\cong l^2(F)\otimes l^2(F\...