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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7
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1answer
278 views

Visualization of SU(3)

I am trying to visualize the $SU(3)$ group used in quantum field theory. I have a (reasonably) good understanding of $SU(2)$ as the double cover of $SO(3)$ and also that this is homeomorphic to $S^3$. ...
2
votes
0answers
38 views

The path-connected field is just $\mathbb{R}$ or $\mathbb{C}$?

A topological ring is a ring $R$ which is also a topological space such that both the addition and the multiplication are continuous as maps. $F$ is a topological field, if $F$ is a topological ring, ...
1
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2answers
36 views

A proper subfield of $\mathbb{R}$ must not be connected?

Let $\mathbb{R}$ denote the real line as a topological field, if $F$ is a proper subfield of $\mathbb{R}$, then $F$ must not be connected?
0
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0answers
10 views

Compact group with dense Kronecker sequence

Let $G$ be a compact abelian group with a dense Kronecker sequence, i.e., a dense sequence of the type $(n{\bf a})_{n\in\mathbb Z}$ for some ${\bf a}\in G$. Can $G$ be made isomorphic to a closed ...
4
votes
1answer
240 views

Question about Quotient space, regarding The left coset space of group $G$ with respect to a subgroup $H$

Let $G$ is a left topological group and $H$ is a subgroup of $G$. Denote by $G/H$ the set of all left cosets $aH$ of $H$ in $G$ (for each $a\in G$), and endow it with the quotient topology with ...
2
votes
1answer
27 views

Is multiplication by $n$ a closed map for an abelian topological group

Let $G$ be an abelian topological group. Let $n$ be an integer and consider $G\to G$ given by sending $x\mapsto nx$. Is this a closed map? I do not have any particular reason to think it should be, ...
0
votes
0answers
58 views

Action of discrete subgroup of Lie Groups

Given $\Gamma$ a discrete subgroup of a lie group $G$ I want to show that the action is wandering: $\forall x\in G\exists U_x\vert \{\gamma\in\Gamma\vert \gamma U_x\cap U_x\neq \emptyset\}$ is finite ...
3
votes
1answer
104 views

Haar measure of quotient group

Suppose $G$ is a (Hausdorff) compact group with normalised Haar measure $\mu$, and that $H\trianglelefteq G$ is a closed normal subgroup. Is it true that the pushforward of $\mu$ to $G/H$ is the ...
0
votes
0answers
20 views

$\hat{H} \cong G/H^{\perp}$?

Let $G$ be a locally compact Hausdorff abelian group, and $H$ a closed subgroup of $G$. Let $\hat{G}$ denote the Pontraygin dual of $G$, i.e. the group of coninuous homomorphisms $G \rightarrow S^1$ ...
1
vote
3answers
72 views

Is there a bi-invariant metric (not Riemannian) on $GL_n$?

Is there a bi-invariant* metric $d$ on $GL_n$ (the group of invertible marices) which generates the standard topology on it? (the subspace topology from $\mathbb{R}^{n^2}$) Note: I mean any metric (...
5
votes
1answer
77 views

Why is the Pontraygin dual a locally compact group?

Let $G$ be a locally compact topological group, and let $\hat{G}$ be the set of continuous characters $G \rightarrow S^1$. We give $\hat{G}$ the topology for which a basis of open sets is $$T(\...
1
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0answers
72 views

Digesting a proof of Bieberbach's theorem on crystallographic groups

Here is a proof of Bieberbach's theorem: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.460.1436&rep=rep1&type=pdf I can follow the computing details, but I have no intuitive sense ...
4
votes
2answers
87 views

how many group structures make $S^1$ a topological group?

let $S^1$ be the subspace of $R^2$ given the usual topology. How many group structures make $S^1$ a topological group?
7
votes
1answer
186 views

Is the action of a finite group always discontinuous?

Let $G$ be a finite group acting on a topological space $X$. Is it true that the action of $G$ is always proper? To say that it is proper we have to show that the map $\theta : G \times X \...
3
votes
0answers
42 views

A problem of a discrete group of smooth isometries acting discontinuously on a smooth manifold.

Suppose that a smooth manifold $M$ is a metric space and that $\Gamma$ is a discrete group of smooth isometries acting discontinuously on $M$. Show that the action is necessarily properly ...
1
vote
1answer
39 views

A specific problem on locally compact topological group Q and non existence of Haar measure

I have recently taken a course on topological groups and their Haar measure and I should first mention I am still a beginner so even though I thought about this for a while, I was hoping someone here ...
2
votes
1answer
66 views

Topology of SL(2,R)

I am interested in visualizing the Lie group $SL(2,\mathbb R)$ topologically. Writing a matrix as $A = \begin{pmatrix} x+w & -y+z\\ y+z& x-w \end {pmatrix}$ I could identify a matrix with ...
3
votes
1answer
23 views

If $G\curvearrowright X$ and $H\leq G$ then $\bar{H}x = \overline{Hx}$

If $G$ is a topological group acting effectively on a topological space $X$ and $H$ is a subgroup of $G$ then is it true that $\overline{H}x = \overline{Hx}$, where $\overline{H}$ is the closure of $H$...
1
vote
1answer
58 views

Compatibility of group structure and topological structure for topological groups

I am fairly new to the concept of topological groups, and would like to understand the underlying idea. My question is about the compatibility between the two structures. The definition of a ...
2
votes
0answers
48 views

Closed subgroups of $Z_{p}^{\times}$

I was able to prove that any closed subgroup of additive group of $Z_{p}$ is of the form $p^{n}Z_{p}$ for some $n$. I asked the same question for the multiplicative group of units in $Z_{p}$, that is $...
0
votes
0answers
16 views

$(\mathbb{C}^n -\{ 0 \}) / \mathbb{Z}_k$ [duplicate]

I would like to know what the space $(\mathbb{C}^n -\{ 0 \}) / \mathbb{Z}_k$ looks like topologically and geometrically, where the group action is given by $z \mapsto e^{\frac{2 \pi i}{k}} z$. For n=1 ...
1
vote
3answers
84 views

The existence of a limit point of a closed set

Walter Rudin define a closed set as: 2.18 (d) A subset E of a metric space is closed if every limit point of E is a point of E. I don't see in this definition nothing about the existence of a ...
1
vote
1answer
85 views

Topology ad Geometry of $\mathbb{C}^n/\mathbb{Z}_k$

I would like to know topologically what the space $(\mathbb{C}^n - \{0\})/\mathbb{Z}_k$ may be thought of as. The paper I am reading says that we let $\mathbb{Z}_k$ act on $\mathbb{C}^n - \{0\}$ via ...
2
votes
1answer
47 views

Discrete VS finite groups

I am having some troubles to understand what the difference is between discrete and finite groups. I know they are defined differently, I can't quite understand the difference. I am guessing every ...
0
votes
2answers
27 views

Is there such thing as an $n$-fold cover of $SO(n)$?

I know already that the $Spin(n)$ group is a double cover of $SO(n)$, but is there such a thing as a triple cover, fourfold or $n$-fold cover? I am interested to know if there are 3-spinors (?) or ...
6
votes
1answer
66 views

Real orthogonal matrices with fixed entry value has zero Haar-measure?

Consider the set of real orthogonal matrices of size $n \times n$ such that one entry, say $a_{i,j}$ for fixed $i$ and $j$, satisfies $a_{i,j}=0$. Has this set zero Haar-measure? A simple proof, in ...
0
votes
0answers
35 views

Classification of symmetric space of non compact type

Is there a classification of rank one symmetric space of non compact type ? Remark, for rank 1, There are three families : 1) hyperbolic n-space, corresponding to the Lie group SO(n,1). 2) complex ...
1
vote
0answers
33 views

Almost periodic compactification with the Gelfand-Naimark theorem

Could anyone please help me with a bibliographic reference presenting the almost periodic compactification of a topological group with the aid of the Gelfand-Naimark theorem? Rudin in "Fourier ...
3
votes
1answer
54 views

Absolute Galois group and continuous homomorphism.

Let $G=\mathrm{Gal}(\bar K/K)$ be the absolute Galois group of a number field $K$. I have some problem to understand that the homomoprhism $$ \rho :G \longrightarrow \mathrm{GL}_r(E)$$ $E$ a finite ...
0
votes
0answers
21 views

What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
0
votes
0answers
57 views

Examples of left-topological compact semigroups

I was reading chapter in Todorčević's book Topics in Topology (LNM 1652, DOI: 10.1007/BFb0096295) which deals with the semigroup $\beta\mathbb N$. Several results about left-topological compact ...
0
votes
0answers
38 views

If a group and a groupoid have the same profinite completions, can anything be said about them?

Pretty much what I ask in the title. Suppose $\mathbb{G}$ is a group and $\mathcal{G}$ is a groupoid. Consider their profinite completions $\overline{\mathbb{G}}$ and $\overline{\mathcal{G}}$. If ...
3
votes
1answer
35 views

Is there an relation between the notions of continuous and discontinuous group actions?

Let $G$ be a group of homeomorphisms of a topological space $X$. The action of $G$ on $X$ is said to be discontinuous at a point $x \in X$ if $G_x :=$ the stabilizer of $x$, is finite. $x$ has an ...
0
votes
1answer
46 views

Continuity using topological groups

I have been give this question. I am just a bit confused where to start? Prove that $g : \mathrm{GL}(n,\mathbb R)\times \mathrm{GL}(n,\mathbb R)\to \mathrm{GL}(n,\mathbb R)$ given by $g(x, y) := xy$ (...
0
votes
1answer
18 views

Compact subgroups of a topological group

Consider $G=(0,\infty )$ , with the metric induced by Euclidean metric from $\mathbb R$ .$G$ is a group under multiplication . Then which subgroups of it are compact $?$ Now $G=\...
2
votes
1answer
32 views

What does the topology on $SL(n, \mathbb{R}) / SL(n, \mathbb{Z})$ intuitively look like?

I have come across Mahler's compactness criterion, and am having trouble wrapping my head around the topology of the moduli space of unit volume lattices. Is there an intuitive way to think about it, ...
3
votes
1answer
29 views

Neighborhood base at the unit element in a topological group

One can define the open set $U$ be such that $U$ is open if for every $x\in U$, $xV\subset U$ for some $V\in\mathcal{N}$. (2) and (3) would give the desired continuity for group multiplication and ...
1
vote
0answers
28 views

Proving the Existence of n-linked knots

I was reading up on knots and links and came across: The Hopf Link: https://en.wikipedia.org/wiki/Hopf_link Solomon's Knot (Double Link): https://en.wikipedia.org/wiki/Solomon%27s_knot Which got me ...
1
vote
1answer
43 views

Topological groups, is continuity a requirement or consequence of this definition?

On the Wikipedia-page for topological groups, it is stated as a requirement that the binary operation and the inverse operation is continuous. In Munkres he makes this definition: A topological ...
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0answers
27 views

Is a locally constant function from a profinite abelian p-group constant modulo an open normal subgroup?

Suppose $f$ is a p-adic valued function from a profinite abelian group $B$ ($Z_{p}$ or $Z_{p}^{*}$) such that for all $x$ in $B$ there is an open set $N(x)$ around $x$ such that $f$ is constant on $N(...
3
votes
1answer
61 views

Fubini's theorem for topological monoids

A topological commutative monoid is a commutative monoid $(X,+)$ with a topology $\cal O$ such that $+$ is a continuous function. There is a notion of infinite sum on topological commutative monoids, ...
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0answers
34 views

How does one visualize elements of standard Iwasawa algebra?

How does one think of elements of standard Iwasawa algebra of a profinite abelian group $B$ ? I mean like elements of $Z_{p}$ are represented as infinite series, is there a way to think of elements of ...
3
votes
1answer
135 views

Is $1+T$ a topological generator for $Z_{p}[[T]]$?

Is $1+T$ a topological generator for $Z_{p}[[T]]$? ($Z_p$ is the ring of p-adic integers)
2
votes
1answer
48 views

Classifying topologically cyclic groups

The group of p-adic integers has a dense cyclic subgroup, i.e. it is topologically cyclic What are some (or all) other non-trivial (i.e. non-discrete) examples of such groups?
2
votes
2answers
94 views

Topological group, a big exercise!

Let $(G,d)$ be a group that is also a compact metric space, and G is topological (The maps $G \times G \to G, (x,y) \to xy$ and $G \to G, x \to x^{-1}$ are continuous). And then we consider $(G,\rho)$,...
2
votes
0answers
43 views

Smooth admissible representations, Hom, tensor and extension of scalars.

Let $G$ be a locally profinite group, and consider $V$ and $W$ smooth admissible representations of $G$ over some field $F$ (or char. $0$). Let $E/F$ be any field extension. I'd like to find ...
11
votes
1answer
143 views

Can there be a Diaconis-Shahshahani Upper Bound Lemma for Compact Groups?

Let $G$ be a finite group and $\nu\in M_p(G)\subset \mathbb{C} G$ a probability measure on $G$ and let $\pi$ be the uniform distribution on $G$. Denote by $d_\rho$ the dimension of a non-trivial ...
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vote
0answers
26 views

Is $(X_G, d_G)$ a compact manifold?

Let $G$ a compact topological group act on $(X,d)$ by isometries. We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It ...
1
vote
1answer
49 views

Definition of Roelcke uniformity

Let $(G,\mathcal T)$ be a topological group. The set of all uniformities on $G$ forms a lattice $\frak A$ and the set of all uniformities on $G$ producing $\cal T$, forms a lattice $\frak B$. The ...
5
votes
1answer
108 views

Is $SO(n)$ a topological space?

I am reading some articles about covering space in Wikipedia. It says that $\operatorname{Spin}(n)$ is the universal cover of $SO(n)$ for $n>2$. I cannot understand how people view groups as ...