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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invariant measure on a quotient of a topological group

Suppose I have a locally compact topological group, $G$, and a closed subgroup $H\leq G$. Suppose $\Delta _G|_H = \Delta_H$ where the $\Delta$ are the modular functions on $G$ and $H$. How can I see ...
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88 views

Mal'cev completion of nilpotent groups

Is the $\mathbb{R}$-Mal'cev completion of a finitely generated torsion free nilpotent group connected and simply connected?. Thanks!
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Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
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43 views

A topological group question about generators

If $G$ is connected topological group and $e \in V$, $V$ is open. Then prove that $V$ is a set of generators for $G$.
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1answer
59 views

Double Coset Closed

Let $G$ be a locally compact group and $H$ a closed subgroup. Under what conditions can we say that the double cosets $H\cdot x \cdot H$ are closed? Is this always true? I am interested mainly in the ...
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Connected subgroups of SU(2) and SU(3)

I am reading 'Lie groups, Lie Algebras, and Representations : An Introduction' by Brian Hall and am unable to do the problem 17 in chapter 3. It says Show that every connected Lie subgroup of ...
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102 views

Finding a certain subsemigroup of $(\mathbb R,+)$ [closed]

Is there a subsemigroup $A$ of $(\mathbb R,+)$ such that it has a limit point and has no intersect with its limit points? Thanks for any hints.
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47 views

Is the following descending sequence nonzero?

Let $K_{1}\supseteq K_{2}\supseteq K_{3}\supseteq \cdots$ be a descending sequence of compact subgroups of compact, torsion-free group $G$. Is $\bigcap_{r=1}^\infty n^r K\neq 0$? (for a positive ...
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31 views

Compact subset of compactly generated group

Let $G$ be a locally compact topological group, that is also Hausdorff and second countable. Let $S$ be a compact subset that generates $G$ as a group, which contains the identity and is closed under ...
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138 views

Proving that a metric space is a group

I'm stuck on this relatively hard problem. Let $G$ be a non-empty set, $d$ a distance on $G$ and $\cdot$ an associative operation on $G$ $\cdot$ is such that $$\forall a \in G , \forall x \in G ...
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1answer
48 views

Question about $S_\infty$ or $Aut(\mathbb{N})$

I have been reading a little Kechris and other random Polish group books, and have come across a question I just can't wrap my mind around. Show that $S_\infty$ or $Aut(\mathbb{N})$ the set of all ...
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142 views

$G$ topological group, $H$ discrete normal subgroup, $p$ projection, form Covering Space.

Let $G$ be a topological group. Let $H$ be a discrete normal subgroup of $G$. Let $p : G \to G/H$ be the projection map. Show that $(G, p, G/H)$ form a covering space. Here is what I have so far: ...
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49 views

Is $SL_1(D)$ toplogically finitely generated, for $D$ a division algebra over a local field?

I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question is finitely generated, so I would appreciate if ...
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1answer
56 views

Finitely generated subgroups of prodiscrete groups

Suppose $(G_n, p_{n+1,n}:G_{n+1}\to G_n)$ is an inverse sequence of discrete groups and (obviously continuous) group homomorphisms. Let $G=\varprojlim G_n$ be the inverse limit (with the usual inverse ...
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1answer
48 views

Compactness in a short exact sequence of topological groups

Suppose $H,G,K$ are abelian Hausdorff topological groups and $0\to H\overset{\alpha}\to G\overset{\beta}\to K\to 0$ an exact sequence of continuous homomorphisms. If $H$ and $K$ are compact, can we ...
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1answer
29 views

Does a Normal Moore paratopological group always have countable chain condition?

Does a Normal Moore paratopological group always have countable chain condition? Or is it separable? Thanks for your help.
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119 views

Open subgroups of $\mathbb{R}$ [duplicate]

Let $G$ be a nonempty open subset of $\mathbb{R}$ (with usual topology on $\mathbb{R}$) such that $x,y\in G$ implies that $x-y\in G$. Show that $G=\mathbb{R}$. Clearly $0\in G$. Now how to show that ...
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Show that if $X$ is discrete, then $\phi$ is continuous.

Let $S(X,X)$ of all mappings of a set $X$ to itself, taken with the topology of pointwise convergence. Define $\phi: S(X,X) \times S(X,X) \to S(X,X)$ with $\phi((f,g))=f \circ g$. Show that if ...
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220 views

Subgroup of the unit circle under complex multiplication

Let $T=\{z:|z|=1 \mathbin{\text{and}} z \in \mathbb C\}$ be the unit circle in the complex plane, considered as a topological group under complex multiplication and the usual topology. Show ...
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1answer
37 views

Show the multiplication mapping of $S \times S \to S$ is not (jointly) continuous

Let $S=R \cup {\alpha}$ be the one-point compactification of the usual space $R$ of real numbers. Define multiplication on $S$ by the rule $xy=x+y$ if $x$ and $y$ are in $R$, and $xy=\alpha$, ...
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2answers
54 views

non-amenable subgroup of an amenable locally compact groups

I begin by recall this two know facts: 1- Every subgroup of a discrete amenable group is amenable 2-Every closed subgroup of a locally compact amenable group is amenable. I need an example of an ...
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99 views

Can a locally compact group with closed singleton be countable but not discrete?

Problem: Prove if a locally compact group $(G,*)$ contains a closed singleton then it must be either discrete or uncountable Proof Given: Assuming $G$ is countable we can write $G = \displaystyle ...
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1answer
222 views

Is there a topological group that is connected but not path-connected?

Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected? If yes: $\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure) ...
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1answer
109 views

In how many ways is $\mathbb R$ a topological group?

This is an easier version of a more general question I proposed, which hasn't received much attention. How many binary operations can we assign to $\mathbb R$ which make it into a group, where group ...
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108 views

How many compatible group structures does a topological space admit?

Suppose I have a topological space $(G,\tau)$ and am interested in whether there exists a topological group $(G,*,\tau)$. In other words, can we assign a binary operation $*$ to $(G,\tau)$ which ...
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316 views

proving that $SO(n)$ is path connected

Our professor gave us exercise to show that $G=SO(n,\mathbb R)$ is path connected. He gave some hints, using them I have come upto this far: I have shown that $SO(n)$ acts on $S^{n-1}$ transitively ...
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A dense subgroup with completion not isomorphic to the big (pro-p) group?

This is an (early) exercise from the book "Analytic Pro-p groups": (p.31, ex. 3(iii)) Give an example of a finitely generated pro-$p$ group $G$ and a dense subgroup $H$ of $G$, with $H$ finitely ...
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1answer
113 views

Profinite completion is complete.

Let $G$ be any group, and $\widehat{G}$ its profinite completion. Is it true that $\widehat{\widehat{G}}=\widehat{G}$, i.e. is it true that $\widehat{G}$ is (canonically isomorphic to) its own ...
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1answer
66 views

problem to show that a map is continuous and open

Consider the family $\hat G$ of all minimal Cauchy filters on a topological group $G$. For every $V\in\mathcal{N}_e$ put $$[V]=\{\mathscr{F}:\mathscr{F}\in\hat G,\;V\in \mathscr{F}\}\;.$$ Denote ...
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Are periodic points dense in the unitary group?

In $U(1) = \{z \in \mathbb{C} : |z| = 1\}$, it is well known and easy to see that the set of $z$ so that $ z^n = 1 $ for some $n \in \mathbb{Z}_+$ are dense. Does this fact generalize to the group ...
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42 views

A Tensor Product Identification

Hi folks just looking at Timmermann's "Introduction to Quantum Groups and Duality" and looking at the algebra of representative functions on a compact topological group $G$. I take off in Example ...
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33 views

Locally compact infinite dihedral group

Is there any topology other than discrete which can be given to an infinite dihedral group to make it locally compact topological group? If no, why is it so?
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43 views

If $X$ is separable then is the group of isometries on $X$ separable?

I'm looking for the conditions on $X$ so that the isometry group of $X$ is separable. We are taking the group to have the operation of composition and the topology of pointwise limits. To be honest, ...
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1answer
59 views

reference request: proof that group characters are a basis for $L^2$

I know the following must be very standard, but I haven't found it in any of the functional analysis books to which I have access. Do you know where I can find a self-contained proof?: If $G$ is a ...
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1answer
44 views

For a metric $d$ on a group $G$, why do $d$ and $d^{-1}$ generate the same topology.

Let $d$ be a metric on a group $G$ and define $d^{-1}$ by $d^{-1}(x,y)=d(x^{-1},y^{-1})$. Why do $d$ and $d^{-1}$ generate the same metric topology on $G$? Let $g \in G$ and $\epsilon >0$. Let ...
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1answer
128 views

Fuchsian groups and topological isomorphism

I have a (finite) presentation of a group and I am wanting to prove that it is not Fuchsian. Because it is given by a presentation, a neat, algebraic description of Fuschian groups would be nice. This ...
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Connectedness of sets acting on topological groups…

I come now with a topological group question. Suppose a topological group $G$ acts on a topological space $X$. Suppose $G$ and $X/G$ are connected. Show $X$ is connected. Me and a few friends have ...
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The minimal divisible extension

For a prime number $p$, $F_{p}$ is the p-adic number groups and $J_{p}$ is the p-adic integer groups. Is $F_{p}$, the minimal divisible extension of $J_{p}$?
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Every neighborhood of identity in a topological group contains the product of a symmetric neighborhood of identity.

Let $(G,\cdot)$ be a topological group and $U$ be a neighborhood of $1$. Then there exists a symmetric neighborhood of $1$, $V^{-1} = V$, such that $V\cdot V \subset U$. Having a hard time proving ...
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Please list a few topological groups that I should learn about.

I'm going through Munkres' Topology book and there's a lot about topological groups. For fear that I'll forget the theorems on them I'd like to connect each thing I prove with a real-world example. ...
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118 views

How do you prove that the complex inverse is continuous?

I tried to show that the continuous at a point $\delta / \epsilon$ definition holds but failed. Now I'm thinking along the lines of multiplicative group: $C \rightarrow C, x \mapsto bx$ has inverse ...
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3answers
281 views

Help proving that complex multiplication is continuous.

Let $(\cdot) : C\times C \rightarrow C, (x,y) \mapsto x\cdot y = xy$, where $C$ is the field of complex numbers. Let $|\cdot|$ denote the standard norm on $C$ and on $C^2$ let the norm be the most ...
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1answer
41 views

Compact open subgroup of a locally compact abelian group

Let $L$ be a compact open subgroup of locally compact abelian group $G$. Is $nL=\{nx;x\in L\}$ an open subgroup of $G$?
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Normalizer of an open subgroup of a compact group

For some reason I cannot figure out the following. Suppose $G$ is a compact, Hausdorff and totally disconnected (but I think compact should suffice) group and $U$ an open subgroup. Why is the ...
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1answer
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To show $V \subset A^{-1}A$ why does it suffice to take $h \in V$ and show $A \cap Ah \neq \emptyset$

I'm reading a theorem on Polish groups... Theorem: Let $G$ be a topological group and $A \subset G$ a nonmeager subset with the Baire property. Then $A^{-1}A$ contains an open neighborhood of $1_G$. ...
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1answer
33 views

Is every discrete, torsion and divisible group $\sigma-$compact?

Let $G$ be a discrete, torsion and divisible group. Is $G$, $\sigma-$compact?
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Let $H$ be a $\sigma-$compact subgroup of locally compact abelian group $G$ such that $G/H$ is $\sigma-$compact. Is $G$ $\sigma-$compact?

Let $H$ be a $\sigma-$compact, closed subgroup of locally compact abelian group $G$ such that $G/H$ is $\sigma-$compact. Is $G$ $\sigma-$compact?
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1answer
42 views

Finding a $V$ open s.t. $V^{-1}V \subset U$

I'm trying to understand a sentence in a theorem about Polish groups. I'll write the proof all the way up to the sentence that I'm having trouble with. Theorem: Let $G,H$ be Polish groups and $\phi: ...
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1answer
29 views

A question on divisible group

Let $G$ be a divisible, locally compact abelian group and $L_{1}\supseteq L_{2}\supseteq L_{3}\supseteq ...$ be a sequence of compactly generated, open subgroups of $G$. Can we deduce that there exist ...
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If a subgroup has finite-intersection with a compact neighbourhood of $1$, then it is discrete?

At page 79 of Basic Number Theory by André Weil, there is an argument showing that a subgroup of a topological group is discrete, because there is a compact neighbourhood of $1$ with finite ...