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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Considering the right Haar measure on the affine group, how does the absolute value come in?

Let $ G $ be the affine group with group action defined by $ (b,a)\cdot(x,s) = (ax+b,as) $ then it is a locally compact group and as such has a Haar measure. In particular the left Haar measure is $ ...
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35 views

$\widehat{\mathbb{T}}$ can be identified with $\mathbb{Z}$

$ \mathbb{T} \stackrel{\text{def}}{=} \{ z \in \mathbb{C} : |z| = 1 \}$ $\widehat{\mathbb{T}} \stackrel{\text{def}}{=} \text{Hom}(\mathbb{T},\mathbb{T})$ To show that $\widehat{\mathbb{T}}$ can be ...
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79 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ ...
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1answer
48 views

Are bounded subsets of Lie groups totally bounded

Let $ G $ be a finite dimensional real Lie group, and take a bounded ball $ B_R(e) \subset G $ in it, coming from the Riemannian metric, which itself is induced from an inner product on $ \mathfrak{g} ...
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755 views

Can $S^2$ be turned into a topological group?

I know that $S^1$ and $S^3$ can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that $S^2$ ...
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2answers
75 views

Is intersection of connected subgroups connected?

Let $G$ be a compact group. If $A_{\alpha}$, $\alpha \in I$ is a family of closed connected subgroups in $G$, then is it true that $\bigcap_{\alpha \in I}A_{\alpha}$ is connected?
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31 views

Summand of a subgroup in a torus

Let $\mathbb{T}^n := \mathbb{R}^n / \mathbb{Z}^n$ denote the $n-$dimensional torus. If $K$ is a closed normal subgroup of $\mathbb{T}^n$, then does there exist a subgroup $L$ of $\mathbb{T}^n$ such ...
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146 views

Measurable function implies equivalent to an exponential function.

This is a follow up to this question. In that question, I answered that an exponential function can be uniquely determined by three properties: a functional equation, a weak continuity assumption, and ...
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1answer
210 views

Profinite topology of a Group

Let $G$ be a group. Consider now the set of all (left for instance) cosets in $G$ of subgroups of finite index. This set is a base for a topology in $G$. I found somewhere that if $G$ is residually ...
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1answer
65 views

A paratopological group with intersection of neighborhoods of a point non-closed

Do you have an example of a (para)topological group $(G,\mathcal T)$ such that $\bigcap_{V\in \mathcal N_1} V$ is not closed? $\mathcal N _1$ is the set of all neighborbood of the identity element of ...
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77 views

Explanation of a passage in Atiyah / Macdonald

On page 105 the authors show that $\hat{\hat{G}} \cong \hat{G}$ (Proposition 10.5) and conclude that the canonical homomorphism $\phi : \hat{G} \to \hat{\hat{G}}$ is an isomorphism. How does the fact ...
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41 views

Continuity of Multiplication by Fixed Element

This is likely a simple question that I'm just missing, but nothing immediately came to mind. When dealing with topological monoids, it is necessary to prove that the group operation is continuous. ...
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2answers
65 views

Introduction to discrete subgroups of the euclidean group

I am looking for a general introduction to discrete subgroups of the euclidean group (= group of isometries in euclidean space). Even though I searched quite a bit, I was unable to find a good ...
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47 views

A homeomorphism of topological groups.

Let $L/K$ be a Galois field extension and $(I, \leq)$ be the directed set of all finite Galois extensions $E$ of $K$ contained in $L$ (we say $E^{\prime}\leq E$ if $E^{\prime}\subseteq E$). If ...
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2answers
63 views

Showing that every path can be well-divided?

Let $\gamma: [0,1] \rightarrow S^1$ be a path. We'll say that $\gamma$ is well-divided if there are $a_1,...a_n$ such that: $a_1=0$, $a_n=1$ $\forall_{1\leq i < n}: a_i<a_{i+1}$ ...
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38 views

Continuous maps of topological groups

Suppose $\theta: G\to K$ is a homomorphism of topological groups. We know that $G$ and $K$ each have fundamental systems of open neighbourhoods of the identity. If I want to show that $\theta$ is ...
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151 views

What topological restrictions are there for a topological space to be a group?

I'm trying to provide a group structure for some Riemannian surfaces. I heard that the following result holds: Let $X$ be a compact Riemannian surface. Then $X$ admits a group structure if, and ...
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1answer
41 views

Haar measure of point sets

Let $G$ be a locally compact group with Haar measure $\mu$ (left or right doesn't matter to me). I know that the Haar measure is positive on open sets. What can be said about the Haar measure on ...
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1answer
38 views

Elements Outside the Identity Component $SO^+(1,\,3)$ of the Lorentz Group $O(1,\,3)$

I have been answering a question on Physics Stack exchange to do with the difference between the "proper orthochronous" (i.e. identity component of) the Lorentz group $SO^+(1,\,3)$ and the Lorentz ...
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2answers
98 views

Closure of a topological group

Let $G$ be a topological group. Define $H(g)=\{g^n\}^{\infty}_{n=-\infty}$ for each $g \in G$. I need to prove that the closure of the set $H(g)$ is a commutative subgroup of $G$. But I am not sure ...
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38 views

Connected component and group action

Let $G$ be a topological group acting on a set $X$. Let $x \in X$ and consider the orbit $G.x$ endowed with the topology coming from the quotient $G/ Stab(x)$. If $G^0$ is the connected component of ...
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1answer
98 views

Topology of $GL_n(K)$

I need to show any of the following results: Consider $K=\mathbb{R}$ or $\mathbb{C}$, then, 1) The compact-open topology and the usual topology of $GL_n(K)$ are the same. 2) Taking inverses and ...
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21 views

Circular jump progrssion

in a circular pond, lotus petals are arranged along the perimeter. a frog leaps from one petal to another in such a way that starting from a petal, it skips one petal and jumps to the next one; then ...
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39 views

A descending filtration on a group defines a topological group.

I'm going through this Wikipedia section on Group Filtrations. Let $G$ be a group and $G_n$ a descending filtration of subgroups ($G_{n+1} \subset G_n$). I've already proved that it forms a ...
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50 views

Showing $U$ open in topological group $G$ $\implies$ $gU$ is open

If $G$ is a topological group, and $U$ is an open set in $G$, then do we have that $gU$ is also open in $G$? I know that since $G$ is a topological group, the mappings $\mu: G^2 \rightarrow G$ s.t. ...
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44 views

Topological Rings and Homothecy

I am trying to show that in a topological ring $A$, if left homothecy $x \mapsto ax$ is continuous at $x=0$ for all $a \in A$ and multiplication $\mu$ is continuous at $(0,0)$, then multiplication is ...
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2answers
87 views

Showing that $\pi(G/H, 1) = H$ under a condition

Problem: Let $G$ be a simply connected (i.e., $\pi(G)=1$) topological group, and let $H$ be a discrete normal subgroup. Prove that $\pi(G/H,1) = H$. I know that since $H$ is a discrete subgroup ...
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90 views

a measurable function on a LCA group coincide with an mulitplicative character almost everywhere

Let $G$ be an LCA group. We say $\tilde \chi$ is a multiplicative character if it is a continuous function : $\tilde \chi : G \to S^1 $ where $ S^1 : = \{ z \in \mathbb C : |z| =1 \}$ is the unit ...
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1answer
159 views

Prove that the Pontryagin dual of a locally compact abelian group is also a locally compact abelian group.

Let $ G $ be a locally compact abelian (LCA) group and $ \widehat{G} $ the Pontryagin dual of $ G $, i.e., the set of all continuous homomorphisms $ G \to \mathbb{R} / \mathbb{Z} $. Clearly, $ ...
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114 views

A question on the classifying space $BG$, its universal property (?), and the stack $[\bullet/G]$

I am learning about the classifying space $BG$ of a topological group $G$. I know the definition $$BG=EG/G,$$ where $EG$ is any contractible space on which $G$ acts freely. If I am not mistaken, with ...
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25 views

Folner condition for locally compact groups?

Let $G$ be a locally compact topological group and denote $\lambda$ the left invariant Haar measure. 1-$G$ satifies the F\o lner condition if for every compact set $F\subseteq G$ and $\varepsilon ...
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1answer
25 views

Connected component of identity of an automorphism group is a subgroup

Let $D$ be a bounded domain in $\mathbb{C}^n$ and let Aut$(D)$ be the set of biholomorphic functions from $D$ to $D$. Define a metric on Aut$(D)$ by the supremum norm, $d(f,g) = \sup_{z\in ...
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61 views

Existence of particular open subgroups, given a profinite group

I have currently read a proof (existence of sections for pro-finite groups (in the book profinite groups of Ribes)) and I did not understand the following two facts used (without mentioning any ...
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1answer
49 views

Existence of a neighbourhood of a compact set ( from james fibrewise topology)

I'm reading James' Fibrewise topology book and I'm trying to understand the proof of proposition 7.4 , it says: Let X be a proper G-space . Then X is fibrewise regular over X/G. Proof For any $x \in ...
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909 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
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53 views

How can we characterize all topological groups given $G$?

The idea is that all topologies on G (not necessarily making it a topo group) can be completely specified by a set of functions $F = \{f: G \to G\}$ if you form a basis for the topology like: $B = ...
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1answer
91 views

Question about solvable cocompact subgroups in linear algebraic group over a finite extension of the p-adic numbers

Let $Q_p$ be the p-adic numbers, where p is any prime number. Then $Q_p$ is a locally compact, Hausdorff, totally disconnected (non-discrete) topological field. Let $GL(n,Q_p)$ be the general linear ...
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Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
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1answer
108 views

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ [duplicate]

Show $\otimes$ and $*$ are the same operation on $\pi_1(G, x_0)$ where $(f\otimes g)(s) = f(s) \cdot g(s)$ where $\cdot$ is the group operation on the topological group $G. $ This is a question from ...
3
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72 views

Integration over uncountable set of characters

Let $G$ be a compact (assumed Hausdorff) group and $\hat{G}$ be the set of all characters of irreducible, finite-dimensional representations of $G$. It might occur that $\hat{G}$ is uncountable. It ...
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1answer
74 views

In what topological abelian groups does convergence to zero imply summability?

Let $\; \langle G\hspace{-0.02 in},\hspace{-0.04 in}+,\hspace{-0.04 in}\mathcal{T}\hspace{.03 in}\rangle \;$ be a $\hspace{.02 in}\big(\hspace{-0.03 in}$$\text{T}_{\hspace{-0.02 in}0}$$\hspace{-0.03 ...
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1answer
18 views

A non-closed $p$-group.

Are there a Hausdorff topological group $(G,\mathcal T)$ and and a non-closed $p$-group $P\le G$ ? a $p$-group where $p$ is a prime number, is a group $P$ such that $$(\forall a\in P)(\exists n\in ...
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101 views

Isometry groups acting transitively

Let $X$ be a metric space and $G$ be its group of isometries. 1) Is it true that $G$ acts on $X$ transitively? If so, where can I find a proof? If not, how can one characterize those $X$ for which ...
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1answer
205 views

Polish topological group

A friend asked me to help him prove that the topological group $\mathrm{Homeo}(0,1)$ (homeomorphism of $(0,1)$ with the compact open topology) is Polish (that is, separable and completely metrizable). ...
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Invariance of Decomposition of Invariant Functional

Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a ...
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2answers
144 views

Intuition behind the failure of unimodularity

If $G$ is a locally compact group then up to normalization it admits a unique Haar measure: a left invariant measure defined on all Borel subsets of $G$, which assigns every compact set a finite ...
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36 views

Probability space associated with a compact group

Is the probability space associated with a compact group with Haar probability always a standard probability space? I recall seeing somewhere the fact that if the topology generating the Borel sigma ...
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2answers
287 views

$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint ...
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1answer
50 views

On homomorphisms of the compact symplectic group

Each $q$ in the symplectic group $Sp(1)$, defines an operator on the imaginary quaternions: $\rho(q):Im\mathbb{H} \rightarrow Im\mathbb{H}$ by $\rho(q)(x)=qx\bar{q}$ We can see here that $\rho:Sp(1) ...
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70 views

What is the cokernel of $\Bbb Q^{\text{disc}} \hookrightarrow \Bbb R$?

These two should be the standard examples for why Locally compact abelian groups are not an abelian categoty. The cokernel of any of these maps should is not a LCAG. $$\Bbb Q^{\text{disc}} ...