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.

learn more… | top users | synonyms

4
votes
0answers
192 views

Discrete Closed Subgroup H of a Simply Connected Topological Group G isomorphic to fundamental group of G / H.

A problem in Rotman's Algebraic Topology is as follows: Given a simply connected topological group G with a closed discete normal subgroup H, show that $\pi_1(G / H) \cong H$. I believe I have this ...
2
votes
1answer
73 views

Uncountable dense measurable subgroups of $\mathbb{C}$

Is it possible to have an uncountable proper dense subgroup of $\mathbb{C}$ which is also Baire or Lebesgue measurable?
1
vote
0answers
48 views

right multiplication by elements of a discrete subgroup preserve left haar measure?

If $\Gamma$ is a discrete subgroup of a locally compact topological group, G, it is not necessarily the case that right multiplication on $G$ by elements of $\Gamma$ will preserve a left Haar ...
2
votes
0answers
162 views

Orientability as a topological property

Can one prove that orientability(of a manifold)is a topological property without using algebraic topology? That is, using a combination of general topology,linear algebra,and topological groups(such ...
2
votes
1answer
69 views

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 ...
2
votes
0answers
218 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!
11
votes
1answer
139 views

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 ...
0
votes
1answer
45 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 $G$.
1
vote
1answer
86 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 ...
3
votes
0answers
286 views

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 ...
5
votes
0answers
109 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.
1
vote
1answer
48 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 ...
2
votes
0answers
47 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 ...
11
votes
1answer
206 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 ...
2
votes
1answer
59 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 ...
2
votes
1answer
403 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: ...
2
votes
1answer
68 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 ...
2
votes
1answer
70 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 ...
2
votes
1answer
71 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 ...
0
votes
1answer
34 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.
2
votes
5answers
160 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 ...
0
votes
1answer
50 views

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 ...
0
votes
2answers
653 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 ...
0
votes
1answer
46 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 $xy=\alpha$, ...
2
votes
2answers
114 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 ...
1
vote
1answer
187 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 ...
13
votes
1answer
381 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) ...
2
votes
1answer
121 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 ...
5
votes
0answers
224 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 ...
3
votes
1answer
718 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 ...
3
votes
0answers
66 views

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 ...
4
votes
1answer
192 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 ...
2
votes
1answer
83 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 ...
3
votes
1answer
68 views

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 ...
0
votes
1answer
58 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 ...
3
votes
1answer
37 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?
0
votes
1answer
59 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, ...
4
votes
1answer
71 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 ...
0
votes
1answer
48 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 ...
3
votes
1answer
197 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 ...
1
vote
1answer
59 views

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 ...
3
votes
0answers
36 views

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 $J_{p}$?
4
votes
2answers
370 views

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 ...
5
votes
3answers
202 views

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. ...
0
votes
2answers
136 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 ...
2
votes
3answers
500 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 ...
3
votes
1answer
55 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 $G$?
1
vote
0answers
90 views

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 ...
1
vote
1answer
24 views

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 $1_G$. ...
1
vote
1answer
45 views

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

Let $G$ be a discrete, torsion and divisible group. Is $G$, $\sigma-$compact?