2
votes
1answer
27 views

What does it mean for a group to act cocompactly by isometries on a topological space $X$?

What does it mean for a group to act cocompactly by isometries on a topological space $X$? I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact. Not ...
1
vote
0answers
44 views

Family of equivalent unitary representations is not a set.

I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ...
5
votes
1answer
41 views

Characters of a Group: two definitions

If $G$ is an abelian group, the characters associated to the rapresentations of $G$ over $\textrm{GL}_1(\mathbb C)=\mathbb C^\ast$ are simply the group homomorphisms: $$\chi:G\longrightarrow\mathbb ...
1
vote
1answer
51 views

Topological group with discrete topology

Let $G$ be a topological group. I came to know that if I can show the existence of a homeomorphism of $G$ which moves only finitely many points of $G$, then $G$ has only discrete topology. How can I ...
0
votes
0answers
29 views

Can a compact topological group have the trivial topology?

I want to show that the following are equivalent for a compact topological group $G$: $G$ is the inverse limit of finite groups $G_i$. There's a family $\left\{N_i\right\}$ of open normal subgroups ...
0
votes
0answers
32 views

Choosing a canonical fundamental domain

I have a set of equations that partitions a certain space into equivalent regions. For a given point $p$ contained in region $R_1$, there are equivalence relations giving its equivalent position in ...
26
votes
5answers
466 views

Can $S^2$ be turned into a topological group?

I know that $S^1$ and $S^3$ can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that $S^2$ ...
1
vote
0answers
26 views

Summand of a subgroup in a torus

Let $\mathbb{T}^n := \mathbb{R}^n / \mathbb{Z}^n$ denote the $n-$dimensional torus. If $K$ is a closed normal subgroup of $\mathbb{T}^n$, then does there exist a subgroup $L$ of $\mathbb{T}^n$ such ...
6
votes
1answer
86 views

Profinite topology of a Group

Let $G$ be a group. Consider now the set of all (left for instance) cosets in $G$ of subgroups of finite index. This set is a base for a topology in $G$. I found somewhere that if $G$ is residually ...
1
vote
2answers
53 views

Introduction to discrete subgroups of the euclidean group

I am looking for a general introduction to discrete subgroups of the euclidean group (= group of isometries in euclidean space). Even though I searched quite a bit, I was unable to find a good ...
2
votes
0answers
35 views

A homeomorphism of topological groups.

Let $L/K$ be a Galois field extension and $(I, \leq)$ be the directed set of all finite Galois extensions $E$ of $K$ contained in $L$ (we say $E^{\prime}\leq E$ if $E^{\prime}\subseteq E$). If ...
0
votes
0answers
33 views

Continuous maps of topological groups

Suppose $\theta: G\to K$ is a homomorphism of topological groups. We know that $G$ and $K$ each have fundamental systems of open neighbourhoods of the identity. If I want to show that $\theta$ is ...
7
votes
2answers
135 views

What topological restrictions are there for a topological space to be a group?

I'm trying to provide a group structure for some Riemannian surfaces. I heard that the following result holds: Let $X$ be a compact Riemannian surface. Then $X$ admits a group structure if, and ...
0
votes
1answer
30 views

Elements Outside the Identity Component $SO^+(1,\,3)$ of the Lorentz Group $O(1,\,3)$

I have been answering a question on Physics Stack exchange to do with the difference between the "proper orthochronous" (i.e. identity component of) the Lorentz group $SO^+(1,\,3)$ and the Lorentz ...
0
votes
1answer
27 views

A descending filtration on a group defines a topological group.

I'm going through this Wikipedia section on Group Filtrations. Let $G$ be a group and $G_n$ a descending filtration of subgroups ($G_{n+1} \subset G_n$). I've already proved that it forms a ...
1
vote
0answers
43 views

How can we characterize all topological groups given $G$?

The idea is that all topologies on G (not necessarily making it a topo group) can be completely specified by a set of functions $F = \{f: G \to G\}$ if you form a basis for the topology like: $B = ...
1
vote
1answer
80 views

Question about solvable cocompact subgroups in linear algebraic group over a finite extension of the p-adic numbers

Let $Q_p$ be the p-adic numbers, where p is any prime number. Then $Q_p$ is a locally compact, Hausdorff, totally disconnected (non-discrete) topological field. Let $GL(n,Q_p)$ be the general linear ...
2
votes
0answers
36 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
2
votes
0answers
41 views

Invariance of Decomposition of Invariant Functional

Let $Q$ a locally compact group acting on a locally compact space $X$ on the left. Let $\mathcal{A}$ a Banach space of bounded continuous functions $f:X\to\mathbb{C}$ and $m\in\mathcal{A}^{\ast}$ a ...
3
votes
2answers
113 views

$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint ...
0
votes
1answer
38 views

Different definitions of topological group [duplicate]

Recently I discovered the definition of topological group. So, topological group is an abstract group $G$ endowed with topological structure such that the maps $mult: G\times G\longrightarrow G$ and ...
11
votes
2answers
153 views

Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
1
vote
1answer
55 views

Constructing Topological Groups [closed]

In general, is there a way to construct topological groups? That is, given two topological groups $X$ and $Y,$ can I construct a topological group $Z$ using $X$ and $Y$ in said construction? I have ...
2
votes
1answer
89 views

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact.

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact. This seems pretty straight forward but how will I be able to prove this? I saw this ...
3
votes
2answers
189 views

What topological group is $\mathbb R/\mathbb Z$?

The integers $\mathbb Z$ are a normal subgroup of $(\mathbb R, +)$. The quotient $\mathbb R/\mathbb Z$ is a familiar topological group; what is it? I've found elsewhere on the internet that it is ...
0
votes
0answers
36 views

center of the group of orthogonal matrices of dim 3 [duplicate]

i am looking for the center of the group of orthogonal matrices of dimension 3. i'm thinking it contains all rotations and reflections but i'm not sure i'm correct and (assuming i am) don't know how ...
1
vote
1answer
53 views

Quotient group $G/G_0$ in Group Topology

I'm stuck on this (apparently) simple thing: If $G$ is a topological group and $G_0$ is the connected component of $G$ containing the identity then $G/G_0$ is discrete if and only if $G_0$ is open. ...
4
votes
1answer
47 views

Is true that $Z(G)/N = Z(G/N)$ for connected topological groups?

Let $G$ be a connected topological group and $N$ a discrete normal subgroup of $G$. Is it true that $Z(G)/N = Z(G/N)$, where $Z(G)$ denotes the center of $G$? I know that every discrete normal ...
2
votes
1answer
54 views

Combining the axioms of a topological group

According to Wikipedia, a topological group $G$ is a topological space and a group, such that the functions $$(x,y) \mapsto x\cdot y\\x\mapsto x^{-1} $$are continuous. Is the single requirement that ...
0
votes
2answers
27 views

H = {A $\in$ G| there exists f:[0,1]$\to$G continuous, such that f(0)=A, f(1)=I}, Is H normal in G?

If G is a subgroup of GL(n;$\mathbb R$) and H = {A $\in$ G| there exists f:[0,1]$\to$G continuous, such that f(0)=A, f(1)=I}, Is H normal in G?
2
votes
2answers
68 views

extension of group operation from $\mathbb{Q}$ to $\mathbb{R}$

I'm having a hard time with this (seems easy, but could be misleading) problem: Let $A \subseteq \mathbb{Q}$ be a convex subset, and let $+$ group operation on $A$. Let $\overline{A} := \{x \in ...
0
votes
0answers
46 views

Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
0
votes
0answers
51 views

Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
3
votes
2answers
71 views

Topological Group $G$ totally disconnected $\Rightarrow$ $G$ hausdorff?

On Wikipedia, I read that a topological group is necessarily Hausdorff if it is totally disconnected. Is that true? I read it on this page: http://en.wikipedia.org/wiki/Totally_disconnected_group ...
1
vote
0answers
42 views

Group operation continuous in the interval topology

I'm trying to prove the following: We have a DLO without endpoints M, and a group operation on M, which is continuous in the interval topology. I want to prove: if $b<c$ then for every $a \in M$ ...
1
vote
0answers
52 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 ...
11
votes
1answer
142 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
51 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 ...
1
vote
1answer
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 ...
4
votes
0answers
109 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
0answers
50 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
117 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 ...
4
votes
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 ...
3
votes
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 ...
1
vote
0answers
66 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 ...
3
votes
1answer
91 views

Upper triangular matrices in $\mathrm{SL}(2,\mathbb{R})$

If $G$ is a compact Hausdorff topological group then every neighborhood of the identity contains a neighborhood $U$ which is invariant under conjugation. That is, $gUg^{-1}=U$ for all $g\in G$. ...
3
votes
1answer
106 views

Counterexample or proof that a certain subset in a topological group is closed

We consider a Hausdorff topological group $G$ acting on a topological space $X$ [action simply means a continuous map $G\times X\rightarrow X$ verifying $(gh)(x)=g(h(x))$, and $1(x)=x$]. The set ...
11
votes
1answer
163 views

Is there a nontrivial topological group that's isomorphic to its fundamental group?

All I know is that the topological group has to be Abelian. I have no idea how to prove or disprove this statement. Thanks in advance.
2
votes
4answers
91 views

Closure of a certain subset in a compact topological group

Suppose that $G$ is a compact Hausdorff topological group and that $g\in G$. Consider the set $A=\{g^n : n=0,1,2,\ldots\}$ and let $\bar{A}$ denote the closure of $A$ in $G$. Is it true that ...
2
votes
0answers
39 views

A certain inverse limit

Let $p$ be an odd prime and $n$ a positive integer. Let $\zeta_{p^{n+1}}$ be a primitive $p^{n+1}$-th root of unity. It can be shown that $Gal(\mathbb Q(\zeta_{p^{n+1}})/\mathbb Q)\cong (\mathbb ...