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

1
vote
1answer
10 views

Effective Topological Transformation Groups and the Group of Homeomorphisms

I'm reading Steenrod's Topology of Fibre Bundles, and on pages 6 and 7, he defines a topological group $G$ and a topological transformation group of a topological space (which I understand to be a ...
1
vote
1answer
56 views

a topological property of the product topology

Let $G$ be a non discrete Polish group. Let $K$ be a compact set of $G$, $C$ a closed set of $G^n$ and $B$ an open set of $G^n$. Suppose $K^n\cap C\subseteq B$. Prove that there is an open set of $G$, ...
7
votes
0answers
89 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 ...
1
vote
1answer
29 views

Describing the clopen sets of a profinite group

I've read somewhere that all clopen subsets of a profinite group $$G \simeq \varprojlim\left(G_i, f_{ij}:G_i \to G_j\right)_{i,j \in I}$$ are exactly the preimages of subsets of the $G_i$'s. It's easy ...
0
votes
1answer
15 views

product of torus and affine space

Given the set $\mathbb{C}^*\times \mathbb{C}$ with the group structure given by $(x,y)\cdot (z,w) =(xz, zy+wx) $. How can I check that this is reductive or not? ? I think that its maximal normal ...
2
votes
1answer
21 views

Regarding part of proof of proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space.

Proposition: Any topological group $(G, \tau)$ which is a $T_1$-space is also a Hausdorff space. Part of Proof: Let $x$ and $y$ be distinct points of $G$. Then $x^{-1}y \neq e$ (identity ...
3
votes
3answers
63 views

Example of Topological Group

I'm trying to learn about topological groups but I can't seem to google anything that provides a simple and clear example of how the set of elements of a group correspond to open sets of a topology. ...
1
vote
0answers
44 views

Some doubts on the relationship between Lie algebras and Lie groups

Let $(\mathbb G,*)$ be a Carnot group. Thus, by definition, $\mathbb G$ is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ admits a stratification, that is $$\mathfrak ...
0
votes
1answer
27 views

The conjugation Group action is continuos

How can I prove that the group action from $G\times G\to G$ defined by $(g,x)\mapsto gxg^{-1}$ is a continuos function? I tried to use the known facts that multiplication and $(x,y)\mapsto xy^{-1}$ ...
0
votes
0answers
31 views

$G_\delta$ subgroups of a Polish Group

Let $X$ be a Polish Group. It's known that every its Polish subgroup is a $G_\delta$. Pick one of them, say $V$. Is it true that $V$ is the intersection of open subgroups? Thank you
0
votes
1answer
37 views

Polish subgroups of $S_\infty$

Let $S_\infty$ considered as Polish Group. Prove that every Polish subgroup of $S_\infty$ has the following form: $\overline{{\left \langle X \right \rangle}}$, where $X$ is a countable subset of ...
2
votes
0answers
28 views

Examples of non unimodular groups

I'm looking for criteria to know if a (lie) group is unimodular, the thing is that I only know one non unimodualr group that arrises naturaly, the affine group ax+b, and their direct generalizations. ...
0
votes
0answers
12 views

is the stabilizer of the connected component of a liegroup contained in the connected component of the stabilizer?

Let $G$ be a non-connected Liegroup acting on a manifold $M$. For $x \in M$ we denote $G_x$ the stabilizer of the $G$-action on $x$. For a arbitrary Liegroup $K$ we denote by $K^\circ$ the connected ...
1
vote
0answers
25 views

Baer-Specker group versus free abelian group generated by an uncountable set [duplicate]

I just learned on Wikipedia that the Baer-Specker group, that is, the group of all integer sequences, is not free abelian. I'm hoping I could be helped to understand why this is true by someone ...
3
votes
1answer
29 views

Topological group of invertible linear transformation??

Suppose $L(\mathbb R^n)$ denotes the set of all invertible linear transformation from $\mathbb R^n$ to itself. It is well known that it is a metric space induced by the norm ...
0
votes
1answer
23 views

Left multiplication isometry?

If $G$ is a semi-simple Lie group and $g\in G$, then $G$ has a bi-invariant metric which is a Riemmanian metric. My question is: with respect to this metric does the left-translation map ...
1
vote
0answers
17 views

Is this a fundamental domain of 2-torus under the action of Z2?

Let $U$ be a vector with relatively prime integer coordinates in $\mathbb{R}^2$. And let $V$ be another vector that is orthogonal to $U$ and the rectangle spanned by $U$ and $V$ are $1$. Is this ...
2
votes
1answer
40 views

Prove that $D^2/\mathbb{Z}_2$ is homeomorphic to $D^2$.

Consider the group of integers modulo 2, $\mathbb{Z}_2 = \{\hat{0},\hat{1}\}$. Define the following action of $\mathbb{Z}_2$ on the closed unit disk $D^2=\{z\in \mathbb{C}: |z|\leq 1\}$: $$ ...
0
votes
0answers
33 views

Topology of a specific shape

How to find topology of this shape? It's Fundamental group, homotopy type and some interesting information about it?
3
votes
1answer
29 views

If $H$ and $\frac GH$ are connected so is $G$

In this proposition: where in the proof, is the closedness of the normal subgroup $H$ used?
4
votes
1answer
56 views

Is every Hausdorff homogeneous space also regular?

Every Hausdorff topological group is regular (completely regular, in fact). Is this true if I replace topological group with homogeneous space? This is not obvious to me because there are Hausdorff ...
3
votes
1answer
66 views

Does there exist a Hausdorff group which is not locally compact?

A topological space is countably compact if every countable open cover has a finite subcover. A topological space $X$ is locally compact if any point has a neighbourhood which is compact. A ...
1
vote
0answers
22 views

Induced representations of compact groups

I am taking a seminar that follows Serre's book "Linear Representations of Finite Groups", and I am preparing a talk on Chapter 7 on induced representations (Frobenius reciprocity, Mackey's formula ...
0
votes
0answers
16 views

Connected components of Lie group stabilizers

I'm trying to show that for $G$ a compact Lie group and $\alpha$ a transitive action of $G$ on a (Hausdorff) connected smooth manifold $X$, if there is $x\in X$ such that the stabilizer $G_x$ is ...
1
vote
1answer
45 views

Complete as a semimetric space but not as a topological group

I shall begin with some definitions. 1) Suppose that $X$ is a topological (additive) group and $(x_{s})\subseteq X$ is a net, we said that $(x_{s})$ is Cauchy whenever $U$ is a neighbourhood of $0$, ...
0
votes
0answers
50 views

Lie algebra of a finite group

I'm trying to find the normalizer of the Pauli group $G_n$ (as a subgroup of $SU(2^n)$) utilizing Lie algebras, as is done in a reference to find the normalizer of the Heisenberg group $HW(n)$. There, ...
0
votes
1answer
37 views

open and closed unit balls in topological vector space

It is a famous fact that in any seminormed space $X$, the open unit ball $B$ and closed unit ball $C$, which are defined by $B=\{x\in X: \|x\|_{X}<1\}$ and $C=\{x\in X: \|x\|_{X}\leq 1\}$ ...
0
votes
0answers
16 views

Analogue of analytic continuation for topological groups

Let $\phi:G\to H$ be a homomorphism of (Hausdorff) topological groups and suppose $G$ is connected. If $ker(G)$ has a nonempty interior, then is it true that $\phi$ is trivial?
4
votes
1answer
57 views

How to find the Haar measure on the group of affine transformations on $\mathbb{R}^n$?

Let $$\operatorname{Aff}_n(\mathbb{R}) := \left\{\begin{pmatrix} A & v\\ 0 & 1 \end{pmatrix}: \, A \in \operatorname{GL}_n(\mathbb{R}), v \in \mathbb{R}^n\right\}$$ be the group of affine ...
0
votes
0answers
10 views

Arrive to the group law in exponential coordinates using the vector fields expressed in exponential coordinates

I need an help with the following question. I have this definition for Engels group $\mathbb E$: it is the only connected and simply connected Lie group that has the Engels algebra $\mathfrak g$ as ...
0
votes
0answers
52 views

Haar measure on $G \rtimes_\phi H$

Let $G, H$ be locally compact, $\sigma$-compact metric groups equipped with left Haar measures $m_G, m_H$ respectively. Let $\Phi: G \times H \to G$ be continuous such that $\phi: H \to Aut(G), \, h ...
0
votes
0answers
20 views

What is a set of topological generators?

On page 54 of this paper, Iwasawa wrote that $S = \{\tau_n^{(a)} \;|\; a \in \mathbb{Z}, a \geq 1\}$ generates the group $U_{n,0} = 1 + \mathfrak{p}_n = 1 + \pi_n \mathfrak{o}_n$ topologically where ...
3
votes
1answer
65 views

Is there a “nice” discontinuous, bijective homomorphism $f: (\mathbb{R},+) \to (\mathbb{R},+)$?

Consider $(\mathbb{R},+)$ as a topological group. Using the axiom of choice, we can construct a $\mathbb{Q}$-basis for $\mathbb{R}$ and using this basis, we can define a discontinuous, bijective ...
0
votes
1answer
14 views

Normal subgroup of Engel group

The Engel algebra $\mathfrak g$ is the Lie algebra generated, as a vector space, by four vectors $X_1,X_2,X_3,X_4$ with the only non trivial commutation relations:$$[X_1,X_2]=X_3, \quad ...
3
votes
4answers
77 views

$H$ is a normal subgroup which is not closed. How to prove the quotient group $G/H$ is not Hausdorff. [duplicate]

$G$ is a Hausdorff topology group, $H$ is a normal subgroup which is not closed. How to prove the quotient group $G/H$ is not Hausdorff. For example, what are the quotient group and quotient topology ...
12
votes
3answers
586 views

Can you give me an example of topological group which is not a Lie group.

I know the definitions of Lie group and topological group are different. Can you give me an example of topological group which is not a Lie group.
1
vote
1answer
21 views

Monodromy action and profinite completion of fundamental group

Let $X$ be a path-connected, locally path-connected and semilocally simply connected space and $x\in X$ an arbitrary base point. Let $p: Y\rightarrow X$ be a connected, finite cover and let ...
0
votes
1answer
21 views

find numbers to define a specific action and quotient

Let $X=[-1,1]\times\mathbb{R}$. Find all numbers $\lambda,a,b\in\mathbb{R}$ with the property that $$n\cdot (x,y):= (\lambda^nx,a+by+\lambda n)$$ defines an action of the additive group ...
0
votes
0answers
36 views

A specific quotient homeomorphic to $S^{n-1}$

Let $\Gamma=\mathbb{R}_{>0}$ be the group of strictly positive reals, endowed with the usual multiplication. Let $X=\mathbb{R}^n\setminus \{0\}$. I already showed that $$ \Gamma \times X ...
0
votes
1answer
28 views

The definition of two-sided uniformity

In the field of uniform spaces, what does it mean to be a two-sided uniformity? Could not find a clear definition of this online.
3
votes
0answers
59 views

Group actions by semi-direct products of groups

I have trouble to understand the second part of the following example which I hope someone can explain to me. First let me explain the initial situation which I feel comfortable with: Consider the ...
3
votes
1answer
45 views

Compact Metric Spaces which are Groups and Topologies

Let $(G,d)$ be a compact metric space which as well is a group. Assume that $(x,y) \mapsto xy$ is continuous as a map $G \times G \to G$ and that group inversion $x \mapsto x^{-1}$ is continuous as a ...
3
votes
1answer
39 views

A pair of non group homeomorphic topological groups

The following assertion is true: If $A$ and $B$ are dense group homeomorphic subgroups of complete Hausdorff topological groups $X$ and $Y$ respectively, then $X$ and $Y$ are group homeomorphic. I ...
2
votes
0answers
26 views

Prove that a subgroup in a Lie group is homogeneous

Let $\mathbb E:=(\mathbb R^4, \cdot)$ be a Carnot group whose Lie algebra is given by $\mathfrak g=V_1\oplus V_2 \oplus V_3$, where $V_1=span\{X_1,X_2\},$ $V_2=span\{X_3\},$ $V_3=span\{X_4\}$, the ...
0
votes
0answers
21 views

Integration on compact group

Let $K$ be a compact topological group, and let $(V,\pi)$ be a continuous representation of $K$ over the complex field $\mathbb{C}$. Denote by $\mathrm{d}$ the Haar measure on $K$. If $v\in V$ ...
2
votes
0answers
80 views

Under what mild condition, inclusion of a conjugate subgroup in the initial subgroup gives equality?

Let $G$ be a group, $H \leq G$ a subgroup, and $x \in G$. If $xHx^{-1} \subset H$, then we may not have $xHx^{-1} = H$. Here is a simple counterexample. But this don't prevent the hope to finding ...
2
votes
1answer
29 views

Open mapping theorem for normed abelian groups

A norm on an abelian group is a function valued in $\mathbb{R}_{\geq 0}$ which satisfies $|x|=0 \Leftrightarrow x=0$, $|{-}x|=|x|$, and $|x+y| \leq |x|+|y|$, not necessarily $|z x| = |z| |x|$ for ...
1
vote
2answers
25 views

Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$.

I am trying to find Example of an infinite non-normal subgroup $H$ of non-abelian (topological)group $G$. For instance, The subset $\mathrm{T}(n,\mathbb{R})\subset \mathrm{GL}(n, \mathbb{R})$ of ...
3
votes
1answer
79 views

Why are the fibers of principal G-bundles homeomorphic to G?

I'm trying to get a grip on the modern geometric formulation of gauge theory, in particular connections on principal G-bundles. However, I am stuck right after the definition already: Virtually all ...
4
votes
1answer
27 views

Find two subgroups of $GL(2,\mathbb{C})$ and an isomorphism between them that is not a homeomorphism.

Find two subgroups $G_1$ and $G_2$ of $GL(2,\mathbb{C})$, and an isomorphism $f:G_1\rightarrow G_2$ which fails to be a homeomorphism. The metric on $GL(2,\mathbb{C})$ is the induced metric from ...