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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What is the algebraic structure of $\Bbb Q_p/\Bbb Z_p$? [on hold]

I am curious about the algebraic structure of $\Bbb Q_p/\Bbb Z_p$. Is there any result in this direction? Thanks!
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4 views

Haar measure, can image of modular function be any subgroup of $(0,\infty)$?

It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind ...
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1answer
24 views

Corollary of the Birkhoff Kakutani Theorem: first countable topological vector spaces

http://planetmath.org/birkhoffkakutanitheorem A topological group $(G,*,e)$ is metrizable if and only if $G$ is Hausdorff and the identity $e$ of $G$ has a countable neighborhood basis. In ...
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51 views

Semi-direct product of groups

The situation is the following: Let be $G$ a locally compact (Hausdorff) group such that $G = H \rtimes_{\alpha} N$ is the semi direct product of locally compact groups $N$ and $H$. Let $A$ be a C$^*$-...
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48 views

Extending the topology on a set to the group it generates

The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form $2^{n_2} \cdot 3^{n_3} \cdot 5^{n_5} \cdot ...$ The tuple ...
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25 views

A quotient by a discrete normal subgroup is locally isomorphic to the group itself

Let $G$ be a connected topological group and let $\Gamma$ be a discrete normal subgroup of $G$. Then why $G$ and $G/\Gamma$ are locally isomorphic?
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1answer
19 views

Definition of $\hat{G_1}\times \hat{G_2}$ where $G_1,G_2$ are abelian groups and $\hat{G}$ is the dual of $G$

The question is in the title. I want to know what happens to $(\chi_G,\chi_H)\in\hat{G}\times \hat{H}$. Are they just passively sitting there as a pair or they give something when applied on $(g,h)$? ...
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27 views

Existence of open subgroup extending a smaller one

Let $G$ be an abelian topological group and $H \subseteq G$ a dense subgroup (equipped with the subset topology). Furthermore let $V \subseteq H$ be a subgroup that is open in $H$. Does there exist a ...
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104 views

Halmos Measure Thoery section 62 exercise 3

Is there a locally compact group $G$ and a Borel measure $\mu$ on $G$ such that \begin{equation*} H=\{g\in G\mid \mu(gE)=\mu(E) \: \text{for all measurable} \: E\} \end{equation*} is not a closed ...
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11 views

Fundamental group of a topological group [duplicate]

Given $(G,\cdot)$ a topological group with identity $e$, is it always true that $\pi_1(G,e)$ is abelian? For what it's worth: I have already shown that if $f,g\in\Omega(X,e)$, then $[f \times g]=[f\...
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1answer
42 views

A problem of nets in topology

Let $G$ be a topological group with neutral element $e$. Let $\pi \colon G \to B(E)$ a (non-continuous) representation of $G$ on a Banach space $E$ by bounded linear operators. Let $T$ an element of ...
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1answer
164 views

Is every topological group the topological fundamental group of an space?

The fundamental group $\pi_{1}(X)$ of a path connected topological space $X$ is the image of $Hom(S^{1},X)$. So the fundamental group can be topologized with quotient topology where $Hom(S^{1},X)$,...
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89 views

Order topology on $\Bbb Q^+$ as a Z-module

Consider the multiplicative group of strictly positive rationals, denoted $\Bbb Q^+$. This can be viewed as a $\Bbb Z$-module, with the primes serving as a basis. If we place the order topology on $\...
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16 views

connected or compact Dirichlet domain

Let $G$ be a second-countable locally compact group and $d$ be a proper (i.e. bounded closed sets are compact) left invariant metric on $G$. Let $\Gamma$ be a lattice subgroup of $G$. Consider the ...
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1answer
30 views

Examples of nilpotent connected locally compact groups which are not Lie groups

I am looking for examples of nilpotent connected (or at least almost connected) locally compact groups which are not Lie groups. Do you know of such examples ?
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1answer
28 views

canonical quotient map on lie group is proper?

Let $G$ be Lie group and $K \subset G$ a compact Lie subgroup of $G$. Let $\pi \colon G \to G/K , \quad g \mapsto g.K=[g]$ denote the canonical projection on the quotient and endow $G/K$ with the ...
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6 views

On volume of arithmetic subgroups

I deal a lot with volumes of arithmetic subgroups, mainly in $SL_2(\mathbf{Z)}$. But I remain not at ease with them, making rough explicit calculations case by case instead of having a general method. ...
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2answers
273 views

If the action of a group $G$ on $\mathbb{R}$ is properly discontinuous then G is isomorph to $\mathbb{Z}$?

Let $G$ be a topological group, acts on a topological space $X$, such that the map $f: G \times X \rightarrow X:(g,x)\mapsto g*x$ is continuous. We say that this action is $properly\;discontinuous$ ...
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1answer
46 views

Sequences $(U_n)$ of neighborhoods of $0$ in a LCA group with $m(U_n)\to 0$

Let $G$ denote an infinite compact abelian group with Haar measure $m$ (so $m(G)=1$). Given a neighborhood $U_1$ of the unit $0$ in $G$ we can find a symmetric neighborhood $U_2$ of $0$ such that $...
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3answers
56 views

Show that the sphere, S, and $\mathbb{R}^2$ is not homeomorphic

I am trying to show that the sphere $S^2$ and $\mathbb{R}^2$ are not homeomorphic.I understand that you can't 'compress' a 3D shape into a 2D plane but I don't know how I would express this formally. ...
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1answer
18 views

Example of non-amenable group which is the inverse limit of amenable groups

1) Does there exist a non-amenable locally compact group $G$ which is the inverse limit $\varprojlim G_i$ of amenable groups $G_i$? 2) Does there exist a non-amenable locally compact group $G$ which ...
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2answers
145 views

Representation theory of locally compact groups

My knowledge about representation theory of locally compact groups is rather scattered. As I got more interested with this subject, I would like to know some good references, where I could learn the ...
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1answer
29 views

A continuous action of a compact group on a uniform space is equicontinuous?

I am stuck on how to prove the statement "A continuous action of a compact group $G$ on a uniform space $X$ is equicontinuous." So, essentially we want to show that for every entourage $\alpha$ of $...
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34 views

Conjugacy classes in topological groups are closed?

EDIT Just realized that this question Conjugacy classes of a compact matrix group is related, but I think that the answer use specific properties of matrix groups, so it doesn't apply. QUESTION ...
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1answer
43 views

Is a translation in a compact Lie group homotopic to the identity?

The following exercise is from Guillemin and Pollack, Differential Topology. Show that the Euler characteristic of the orthogonal group (or any compact Lie group for that matter) is zero. Hint: ...
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1answer
126 views

Isomorphism between $SU(2)$ and $U(1, \mathbb H)$

Question: Prove $SU(2)$ is isomorphic to the group of quaternions of norm $1$, that is, $U(1,\mathbb H) \simeq SU(2)$. Attempt: How could I start finding the isomorphism? Intuitively, a ...
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2answers
38 views

Is the product of closed subgroups in topological group closed?

Just out of curiosity: If $G$ is a topological group and $H, K$ are closed subgroups, is $H\cdot K$ a closed subgroup? Thanks!
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15 views

Showing some transformation is group isomorphism (topological group).

Let's Mg be a set of all real valued functions defined on topological group G. Assume that $f:G \to R$. Let's $a \in G$, then define $f_a(x):=f(ax)$ for all $x \in G$. Now define $h_a(f):=f_a$ we know ...
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186 views

Which Algebraic Properties Distinguish Lie Groups from Abstract Groups?

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group, and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
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35 views

A dense set in $\mathbb{T}$

Let $$\mathbb{T}=\{ z \in \mathbb{C}: |z|=1\} $$ Consider $\mathbb{T}$ as a topological group under multiplication with it's usual topology, I'm reading a proof wich states that a dense set in $T$ ...
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23 views

Euclidean Sphere

Consider the Euclidean sphere $S^n = \{x\in \mathbb{R}^{n+1}: ||x||_2=1\}$ of dimension $n\ge1$. Show that for every continuous function $f:S^n\longrightarrow \mathbb{R}$, there exists $x\in S^n$ such ...
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0answers
57 views

Isomorphism theorems for topological groups

I know that the second isomorphism theorem for groups doesn't hold for topological groups, the version that I have for the second isomorphism theorem is: If $G$ is a group, $H$ a subgroup of $G$ and ...
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50 views

Show that $S^1$ acts on $S^3$

$S^3=\{(z_1, z_2) \in \mathbb{C^2} \mid |z_1|^2 + |z_2|^2 = 1 \}$ Show that $S^1$ acts on $S^3$ by $z \cdot (z_1, z_2)=(zz_1, zz_2)$ An action of a topological group $G$ on a topological ...
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41 views

identity for quaternions' group Sp(n)=Sp(2n,C)∩U(2n)

Could you help me for solving this: Let $Sp(n)$ be the group of linear transformations of $H^n$ such that preserve hermitian form $$\sum_{i=1}^n \overline{q_i}r_i,$$ that $H$ is the quaternions _ the ...
4
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1answer
221 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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3answers
29 views

Proving that $B:=\{f(x)\in C[a,b]:f(a)=0\}$ is close set

Let $B$ be a group of all the continuous functions in the interval $[a,b]$ such that $f(a)=0$. Prove that $A$ is close group in the metric space $C[a,b]$ My attempt: Metric space $C[a,b]$ defined ...
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70 views

Fundamental groups of path connected subspaces

Does every path connected subspace of $\mathbb{R}^2$ have a fundamental group the trivial or an infinity group? For example, for convex subspaces we know that, but if we take only path connected ...
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1answer
111 views

General Linear Group over the quaternions is a topological group

How to show that General Linear Group over the quaternions is a a topological group?
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20 views

Topology of bundle maps in Atiyah-Singer IV

I'm trying to read "The index of elliptic operators IV" by Atiyah and Singer, and I do not understand why the topology on $\mathrm{Diff}(X,E)$ on page 123 is constructed in such a peculiar way. Is ...
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1answer
8 views

If $f(gK) \subset U$ find a compact neighborhood of $g$, $\overline{Q}$, such that $f(\overline{Q} K) \subset U$

sorry for my bad english. I am in a proof and I get stuck in the following step. Let $f : X \to Y$ be a continuous function between two topological spaces, $G$ a locally compact topological group, ...
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1answer
28 views

$T_0$ is equivalent to $T_1$ in a topological group

I need to prove that the separation axiom $T_0$ is equivalent to $T_1$ in a topological group $G$. $T_1$ implies $T_0$. So I only need to prove $T_0$ inplies $T_1$. What I have tried is: Suppose $...
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92 views

A $T_0$ topological groups is $T_{3.5}$ (and consequently $T_3$)

I don't understand: why is this so? I've just seen the proof that a $T_0$ topological group is $T_1$, but don't know how to show that it's $T_{3.5}$. BTW, the fact that $x\overline{V}=\overline{xV}$, ...
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1answer
19 views

Lie group structures inside of Clifford algebras

I am reading a text by Jean Gallier on Clifford algebras, Pin and Spin groups. I have a problem with one little innocent-looking paragraph establishing Pin and Spin as Lie groups on the page 37. I don'...
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25 views

Closed subgroup implies open?

If $H$ is a closed subgroup of a topological group, $H$ is also open?, I know that an open subgroup of a topological group is also closed, but the converse is true? if isn't, wich could be a ...
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26 views

Show that $ \operatorname{Sp}(n)=\{A \in M_n(\mathbb{H}) \mid AA^*=I=A^*A\} $ is a compact group

Let $M_n(\mathbb{H})$ be the set of all $n \times n$ matrices with entries in the quaternions $\mathbb{H}$. For $A=(a_{ij} ) $ let $ A^*=(a^*_{ij} ) $ be the matrix with $a^*_{ij}=\bar{a}_{ij} $, ...
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2answers
472 views

Haar Measure of a Topological Ring

A topological ring is a (not necessarily unital) ring $(R,+,\cdot)$ equipped with a topology $\mathcal{T}$ such that, with respect to $\mathcal{T}$, both $(R,+)$ is a topological group and $\cdot:R\...
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1answer
34 views

Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Let $G$ ge a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogenuous space $\frac{G}{H}$ are connected, then $G$ is connected. Remark: For this proof I use the following ...
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15 views

Binary operations being continuous under a topology?

For a set $S$ and some function $f: S \rightarrow S$ and $a\in S$, $f$ is continuous at $a$ under a topology $N$ if for all neighborhoods $N_1(f(a))$ there exists a neighborhood $N_2(a)$ such that $f(...
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1answer
24 views

Proving that Lie groups are locally connected

I'm trying to show that if $G$ is a Lie group, then it is locally connected, i.e., for any point $p$ in an open subset $U$ of $G$, there is a connected neighborhood $V$ of $p$ such that $V\subset U$. ...
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26 views

Show that $\|\cdot\|_Q$ is a seminorm on $l_{\infty}$

On the space $l^{\infty}$, define $||.||_Q=\limsup |x_n|,$ $x=(x_n)$ belongs to $l^{\infty}.$ Show that $||.||_Q$ is a seminorm on $l^{\infty}$ and that its null space is precisely $c_0$, where $c_0$ ...