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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If G is a compact semisimple Lie group and Z is its center, is G/Z always compact?

The title pretty much sums up the question: Suppose $G$ is a compact semisimple Lie group with center $Z$, The question is if $G/Z$ is always compact? or, under which conditions will it be compact?
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1answer
124 views

Why is SO(3) not $S^1 \times S^2$? (Where is the mistake?)

I was trying to calculate the fundamental group of SO(3). In order to represent the group I reasoned the following way: In order to build the 3X3 orthogonal matrix I need an orthonormal positive ...
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26 views
+50

Do the maps need to be surjective in the definition of a profinite group?

Let $G_i, f_{ij}$ be an inverse system of topological groups where each $G_i$ is finite in the discrete topology. A profinite group is defined to be an inverse limit of such an inverse system. ...
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52 views

Why is topological group not a popular topic?

In Japan, there are many universities with a formal course about topological group using the classic by Pontryagin. Yet topological group is not studied in a formal course in many other countries, ...
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1answer
21 views

a generalization of punctured cylinder

Let $S^1\times \mathbb{R}$ be the infinite cylinder. Pucture it, we have $S^1\times \mathbb{R}-*$. Then $(S^1\times \mathbb{R}-*)\simeq Skeleton^1(S^1\times \mathbb{R})\simeq S^1\vee S^1$. How ...
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25 views

Question regarding characters and point open topology

I was wondering why the following claim is correct: Let G* be the group of all continuous homomorphisms from the topological group G and the unit circle (call it T). Then G* is an intersection of a ...
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1answer
44 views

Groups with no nontrivial topology

Does there exist a group $G$ such that $G$ has no topology on it such that $G$ is a topological group apart from the (in)discrete topology (or other such trivalish topologies)? I am asking as ...
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2answers
27 views

Covering groups of compact connected Lie groups is compact.

Hi: I have a question as follows: Is it true that the covering group of a compact, connected Lie group is also compact? Thanks very much!
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2answers
30 views

Normal subgroup of a compact topological group

Is a normal subgroup of a compact topological group closed? What if the group is not compact?
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1answer
37 views

When does a topological group embed topologically in its group of homeomorphisms?

Let $X$ be a topological group. $X$ acts freely on itself by left multiplication; this gives us an injective group homomorphism $\Phi: X\rightarrow \operatorname{Homeo} X$. Under what conditions is ...
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0answers
102 views

Harmonic Analysis on the Affine Group

In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking ...
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0answers
62 views

Exponential map is surjective for compact connected Lie group

How do I show that for every compact connected group $G$, the exponential map $\exp \colon\mathfrak{g} \rightarrow G$ is surjective? I tried to find the proof on the internet but most of them are ...
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1answer
50 views

Proof that difference of compact and closed sets is also closed [duplicate]

I am trying to prove that for any $A$ compact, $B$ closed sets $\Rightarrow A-B = \{a-b | a\in A, b\in B\}$ is also closed, where A and B are subsets of a topological vector space $X$.
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1answer
35 views

Subgroups of the reals of finite index.

I'm studying profinite groups, and I know that every group can be endowed with the profinite topology, whose basis consists of the cosets of subgrous with finite index. Moreover this topology is ...
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1answer
36 views

The group of continuous homomorphisms

Consider the topological group $\Bbb Q$ with the subspace topology and $\Bbb Z$ with the discrete topology. Is there a characterization for $C^{1}(\Bbb Q,\Bbb Z)$, the group of continuous ...
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1answer
46 views

Group structures on Hausdorff space

Could anyone give me some practical (and possibly intuitive) examples of Group structures on Hausdorff spaces? Let us say you had to get freshmen university students interested into fields of maths ...
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1answer
73 views

Compactness and completeness in groups

I know that, in metric spaces, compactness implies completeness. In fact, (i) compactness is equivalent to the fact (ii) every infinite set has an accumulation point and to the fact that (iii) any ...
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0answers
60 views

A counter example

If a set is compact in $Z(\mathbb{A})\setminus GL(2,\mathbb{A})$,then can it be compact in $GL(2,\mathbb{A})$ ? ($\mathbb{A}$, is the adele ring of $F$ on which $GL(2)$ is and $Z$ is the center of ...
2
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1answer
38 views

question regarding closure of symmetric neighborhood of e

I was going through the open mapping theorem (for topological groups) when I stumbled upon a topological property that I couldn't prove to myself. If I have a topological group, $G$ which is ...
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0answers
16 views

Is every closed subgroup of dual group an annihilator?

This is a naive question as I am new to topological group theory. Let $G$ be a Locally Compact Abelian group (LCA group). I know that to every closed subgroup $H$ of $G$ correspond a closed subgroup ...
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1answer
24 views

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ becomes a LCA group?

How can we topologize the character group $\Gamma$ of a locally compact abelian group $G$, such that $\Gamma$ itself becomes a LCA group ? I would really really appreciate if I can get a step by step ...
2
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1answer
72 views

An example of finite, connected topological group

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think ...
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38 views

Proof that a particular subgroup is proper

I've been stuck on this for a long time ... I'm reading a textbook which simply states "this subgroup is proper" but it doesn't make sense to me. Context: I have a pro-$p$ group $G$, which just means ...
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1answer
30 views

In the quotient $G/H$, why we must suppose that $H$ is closed?

We have the following known statement: Theorem: If $G$ is a topological group and $H\subseteq G$ is a closed invariant subgroup of $G$, then $G/H$ (of course with the quotient topology) is a ...
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39 views

Question about Modular function in Haar measure

I'm reading the book "Basic Lie Theory" (http://guests.mpim-bonn.mpg.de/abbaspou/Lie-Book_verrouille.pdf) and I'm trying to understand the proof of Lemma 2.3.4 which states that: Let $G$ be a locally ...
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184 views

On infinite groups admitting finitely many group topologies

It has been proved there is an infinite group which admits exactly two group topologies [1]. For which $n$, is there an infinite group $G$ which admits exactly $n$ group topologies ordered linearly ...
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2answers
79 views

A proof for $\widehat{\Bbb Z_{p^\infty}}\cong Z_p$

According to wikipedia, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers. Where can I find a proof for it on the internet?
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11 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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1answer
43 views

Error in Cariolaro's Unified Signal Theory

From what I understand, in the category $\mathsf {LCA}$ of lca groups, isomorphisms should respect both topology and group structure, hence they are continuous homomorphisms. I'm trying to learn ...
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0answers
28 views

Intersection of invariant subsets of a local group action

I don't understand some facts about invariant subsets of a local group action. Basically (to save you reading definitions) local actions are germs of partial actions which in turn are just like ...
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0answers
59 views

Slice at a point of a topological space

The definition is from the following link -Slice at a point of a topological space Let $G$ be a topological transformation group of a Hausdorff space $X$. A subspace $S$ of is called a slice at a ...
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2answers
31 views

Smallest open, dense, G-invariant subset of a metric space

Let $X$ be a metric space and $G$ be a topological group acting continuously on $X$. Let $ \mathcal S $ be the set of open, dense and $G$-invariant subsets of $X$. I need to take inverse limit (of ...
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1answer
21 views

Weyl Groups/Borel

Could someone tell me where to find a proof of the following statement that I found in some notes about characteristic classes I was reading? If $G$ is a compact connected Lie group with maximal ...
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0answers
16 views

A compact group with a finite dimensional faithful representation [duplicate]

Theorem: If $G$ a compact group has a finite dimensional faithful representation $W$, then any irreducible representation $V$ is contained in $W(k,l) = W^{\otimes k} \otimes (W^*)^{\otimes l}$ for ...
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1answer
53 views

The natural isomorphism in the Pontryagin Duality

Pontryagin duality is the statement that there's a natural isomorphism between the identity functor on $\mathsf{LCA}$, the category of locally compact (Hausdorff) abelian groups with continuous ...
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1answer
57 views

Prove that $SL(n,R)$ is connected.

Prove that $SL(n,R)$ is connected. The problem is I know only topological groups from Munkres only. Again Just started fundamental groups. So if anyone can explain me how it is true in a lucid ...
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73 views

Prove that the quotient map $P:G \to G/H$ is a covering space.

Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space. My ...
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1answer
58 views

Why is the character group defined as $\mathsf{Hom}(G,\mathbb T)$, i.e why is the codomain specifically $\mathbb T$?

In the paper Category Theory Applied to Pontryagin Duality by Roeder, the character group of an lca group is defined as the topological (under the compact-open topology) abelian group of continuous ...
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2answers
47 views

Show that every topological group is $T_3$

I know that it is sufficient to show that for one point ($e$) and any neighbourhood $U$ of $e$, we have a neighbouhood $V$ with $\bar{V} \subseteq U$. Since $x \to x^{-1}$ is continuous, it follows ...
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33 views

Cech-complete separable groups

It is well known that a Baire measurable homomorphism between Polish groups is continuous. Is the same true if we replace Polish groups by Cech-complete, separable groups? It is not true ...
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25 views

What's the significance to the $m$ in the notation $L(n,m)$ for the Lens space?

I'm reading a quick example (Example 12.13 of Topological Manifolds by John Lee) of the construction of the lens space $L(n,m)$. Basically, let $$S^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\}$$ ...
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41 views

Proving that any continuous homomorphism of $\mathbb{R}/(2\pi\mathbb{Z})$ int0 $T$* is neccesarily an exponential function

This is an exercise form Katznelson's book on Harmonic Analysis, so I want to solve it using his hint. T* here denotes the multiplicative group of units of complex numbers of unit norm. That is to ...
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1answer
70 views

Clarification of notion of proper group action.

In a course on differential manifolds and Lie groups, the following theorem was stated, though never proven: Let $M$ and $N$ be smooth manifolds, and suppose $G$ is a Lie group acting on $M$. If ...
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1answer
42 views

question on subgroup of compact group

Suppose $G$ is a compact metrizable abelian group, is it true that $G$ has no finite index subgroups iff $G$ is connected? Any reference or help is appreciated. Thanks in advance! Here are my ...
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267 views

Does every homogeneous space allow a group structure?

Let $(X,\tau)$ be a homogeneous space, that is for all $x,y \in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there a group operation $*:X\times X\to X$ such that ...
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1answer
52 views

Counterexample of “the product of open subsets is open in a topological ring”?

Given a topological ring $R$ and $U,V$ open subsets, we can show that $U+V$ is an open subset due to the fact that $x\mapsto x+y$ is a homeomorphism for every $y \in R$. Since, in general, $R$ is not ...
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1answer
69 views

Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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2answers
72 views

if $\pi_1(G)$ is trivial, how to prove $ \pi_1(G/H)=\pi_0(H)/\pi_0(G) $?

Let $G$ be a topological group, $H$ be its normal subgroup of $G$, and $G/H$ be the quotient space induced by the natural map.(We know that $G/H$ is again a topological group) If $\pi_1(G)$ is ...
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0answers
25 views

cohomology of unordered configuration spaces of sphere

Let $F(X,n)$ be the configuration space of order $n$. Let $F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. What is $H^*(F(S^2,n)/\Sigma_n;\mathbb{Z}_2)$? I did not find the answer ...
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1answer
45 views

Translation invariant metrics and topological groups

During a lecture was pointed out that one of the main feature, from a topological perspective, of normed vector spaces is the translational invariance, that is that one can study the topological ...