# Tagged Questions

A topological group is a group $G$ together with a topology on the elements of $G$ such that the group operation and group inverse function are both continuous (with respect to the topology).

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### Normal subgroups of Aut(M)

Let $S$ be the set of all finite permutations of $\mathbb{N}$, i.e. they fix all but a finite set, and $A\subset S$ the set of all even permutations. Theorem The normal subgroups of $S_\infty$ are ...
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### Automorphism group of a structure without the SAP

A few years ago, a number of examples were given of Fraisse structures without the SAP in answer to the question raised in A Fraïssé class without the strong amalgamation property. It is ...
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### Group actions on principal groupoids

Suppose that $\mathcal{G}$ is etale principal groupoid and that $G$ is a discrete (or finite) group acting freely on the locally compact unit space $\mathcal{G}^0$ (or assuming compactness, if ...
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### LCH topologies on Groups that are not group topologies

Ellis's 1957 paper on Locally Compact Transformation groups proves the following: A locally compact hausdorff topology on a group $(G, \cdot)$ for which left and right multiplication are (separately)...
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### A question about Raikov complete topological groups

I asked this yesterday on Math Stackexchange, but didn't get any comments. So I thought I might ask here too. Let $G$ be a topological group and let $G^*$ be its Raikov completion, i.e its completion ...
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### Set product of profinite subgroups of a compact group is profinite

Let G be a compact group, suppose $G=AB$ where $A$,$B$ are profinite subgroups of $G$. Is it true that G is profinite group?
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### A possible mistake in Walter Rudin, “Fourier analysis on groups”

I have the following lemma 4.2.4 on page 80 in the book (we have locally compact abelian topological groups $G_1, G_2$ and their duals $\Gamma_1, \Gamma_2$): Suppose $E$ is a coset in $\Gamma_2$ ...
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### Centralizer of a dense subgroup in a maximal subgroup of a reductive group

I am looking for a reference to the following statement "Let $G$ be a reductive algebraic group and $K$ a maximal compact subgroup of $G$. If $H$ is a dense subgroup in $K$, then the centralizer of $H$...
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### A non locally compact group of finite topological dimension?

Is there a topological group which is Hausdorff, first countable, locally connected and has finite topological dimension, yet fails to be locally compact?
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### Can approximately periodic functions be perturbed to periodic functions on a locally compact group?

Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...
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### Is $1+T$ a topological generator for $Z_{p}[[T]]$? [closed]

Consider the ring of formal power series $\mathbb{Z}_p[[T]]$ (where $\mathbb{Z}_p$ denotes the ring of $p$-adic integers) with the topology in which a neighborhood basis for $0$ is given by the ideals ...
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### In a topological group $G$ with its lower uniformity, if $G$ is locally totally bounded, is its completion locally compact?

There has been work done on groups whose lower uniformity (or Roelcke uniformity) is totally bounded, e.g. the orthogonal group on a Hilbert space. This condition is equivalent to saying the lower ...
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### Bohr compactification and “discretization”

Let $G$ is a compact group. We can form the Pontriagin dual $\widehat{G}$ of $G$: it is then discrete space. One can consider the Bohr compactification $b\widehat{G}$ of $\widehat{G}$ which is compact ...