# Tagged Questions

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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### 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 $S^3$. ...
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### 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, ...
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### 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?
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### $\hat{H} \cong G/H^{\perp}$?

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 $G \rightarrow S^1$ ...
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### 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 (...
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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(\... 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 ... 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? 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 \... 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 ... 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 ... 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 ... 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... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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{... 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 ... 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 ... 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 ... 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 (... 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=\... 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, ... 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 ... 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 ... 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 ... 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(... 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, ... 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 ... 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) 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? 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),... 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 ... 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 ... 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 ...
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 ...
### 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 ...