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Questions tagged [topological-groups]

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.

A topological group is a group endowed with a topology such that both the group operation and inversion are continuous maps. Every group can be understood as a topological group, if we take the discrete topology.

Topological groups are useful in various areas of mathematics. Every topological vector space is a topological group. Locally compact groups are important in harmonic analysis, see e.g. Pontryagin duality.

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Topological groups - Actions and Hausdorffness

I guess this problem is widely-known, but I couldn't finish it. If $X$ is a topological group (and compact), and $G$ a closed subgroup acting on $X$ by left translation, show that $X/G$ is Hausdorff. I was trying to use that famous lemma about an…
Br09
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Is "$ax+b$" group amenable?

Is the so-called "$ax+b$ group" (the group of affine maps on $\mathbb R$) amenable? One reprersentation for $ax+b$ group is the set of 2 by 2 matrices for form $$\begin{bmatrix}x & y \\0 & 1\end{bmatrix};\,\,\,x,y\in\mathbb R, x>0$$ endowed with…
BigM
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topological isomorphism between a topological group and the identity component of a topological group

Let $\Bbb R$ be the group of real numbers with the usual topology and $\Bbb Z$ the group of integers with the discrete topology. Is $\Bbb R$ topological isomorphism by the identity component of $(\Bbb R\bigoplus \Bbb Z)/H$ where $H$ is the subgroup…
Aliakbar
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The whole group is covered by compact translating of subgroups

$G$ is a locally compact (may not necessarily Hausdorff) group, $H$ is a subgroup in $G$, $G/H$ is compact as a quotient space , then there exist a compact subset $K$ such that $G=KH$(or $G=HK$).
David Chan
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Normal subgroup of a compact topological group

Is a normal subgroup of a compact topological group closed? What if the group is not compact?
WWK
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How the finite cyclic group $\Bbb{Z}_p$ can be endowed with discrete topology to make it a topological group?

How the finite cyclic group $\Bbb{Z}_p$ can be endowed with discrete topology to make it a topological group? We have information that in discrete topology all subsets of $\Bbb{Z}_p$ is open set and it is the largest topology on $\Bbb{Z}_p$.
MAS
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Continuous homomorphism into locally compact Hausdorff group

Could any one give me hint to solve this one? $f:G\rightarrow H$ is continuous homomorphism into a locally compact Hausdorff group $H$. Then we need to show $f$ is necessarily open. all spaces 2nd countable, hausdorff and locally compact
Myshkin
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