5
votes
0answers
155 views

Distributivity of group topologies on $\Bbb Z$

Let $\mathcal L$ be the set of all group topologies on $\Bbb Z$. It is known that $(\mathcal L,\subseteq)$ is a modular complete lattice. Is $(\mathcal L,\subseteq)$ distributive?
7
votes
1answer
399 views

Group structure on an arbitrary completely regular topological space that makes $(x,y)\mapsto xy^{-1}$ continuous at $(1,1)$

Let $(G,\mathcal T)$ be a completely regular topological space. Is there a group structure on $G$ such that the function $$f:G\times G\to G$$ $$f(x,y)=xy^{-1}$$ is continuous at $(1,1)$?
4
votes
1answer
165 views

Are infinite groups “locally topologizable”?

Does every infinite group admit a Hausdorff topology such that the multiplication and inverse are continuous at $1$ but $1$ is not an isolated point? The question is inspired by and related to ...
4
votes
1answer
110 views

Equations and random subgroups in compact groups

EDIT: Here is a more specific question. Let $G$ be a compact group and let $w$ be a word in $d$ variables. Then the solution set $S$ of the equation of $w=1$ is a closed subset of the product $G^d$ ...
3
votes
0answers
52 views

Separation of topological group elements by invariant neighbourhooods

Let $G$ be a topological group that is Hausdorff, that is, for every pair $(g,h)$ of distinct elements of $G$, there exist disjoint open sets $U_g$ and $U_h$ such that $g \in U_g$ and $h \in U_h$. ...
6
votes
2answers
218 views

Union of conjugates of a closed subgroup of a compact group

Let $G$ be a compact Hausdorff group, $H \leq G$ a closed subgroup of infinite index in $G$. Is it possible that the conjugates of $H$ cover some open neighbourhood of $1$ in $G$ (or the whole of ...
0
votes
0answers
45 views

The set of (property) elements of a locally compact group is closed

For which properties $(P)$ is the following statement known to be true? In any locally compact group $G$, the elements of $G$ that satisfy $(P)$ form a closed subset of $G$. In other words, the ...
2
votes
0answers
79 views

Free profinite completions

Let $m,n \in \mathbb{N}$. Which residually finite groups $G$ generated by $m$ elements, have the free profinite group on $n$ generators as their profinite completion?
3
votes
0answers
92 views

Automorphisms of profinite groups

Let $d,n \in \mathbb{N}$, and $p$ a prime number. Let $F$ be a free pro-$p$ group on $d$ generators. Is there an automorphism of $F$ of order $n$?
4
votes
3answers
523 views

Action of a profinite group

Let $G$ be a finitely generated profinite group, $p$ a prime number. Put $$ V = \prod_{i \in I} \mathbb{Z}_p$$ a (profinite) group equipped with the product topology (for convenience, $I$ may be ...
6
votes
1answer
207 views

Topological groups defined by completely disconnected subgroups

Can you define a group topology on a group by specifying which subgroups should be discrete with respect to that topology (where a subgroup $S$ of $G$ is discrete if each $s\in S$ has an open ...
5
votes
1answer
262 views

Locally finite compact groups

I assume all tolpological groups here to be Hausdorff. A group is called locally finite if every finitely generated subgroup is finite. What can be said about a locally finite compact group? Must it ...
0
votes
1answer
80 views

Is a weakly separable group always Lindelöf?

By "weakly separable" I mean the notion for uniform spaces used by David Wigner and Lawrence Brown: a uniform space is weakly separable if any uniform cover has a countable subcover. For a topological ...
15
votes
1answer
474 views

Is a left topological group which is a manifold a topological group?

Let $G$ be a left topological group, i.e. a topological space with group operation such that left multiplication $L_g : x \mapsto gx$ is continuous (but right multiplication and inversion are not ...
2
votes
2answers
298 views

Is any continuous group homomorphism from R to C* an exponential map? [closed]

Consider $\mathbb{R}$ to be an additive topological group, and $\mathbb{C}^{\ast}$ to be a multiplicative topological group. Is the following statement true? If so, then how can one prove it? ...
11
votes
2answers
384 views

subsets of groups which have to be closed no matter what

One example of a subset of a group $G$ which has to be closed in any topology on $G$ compatible with the group operations is a centraliser. Are there any other interesting examples?
6
votes
1answer
150 views

Do syndetic sets on amenable semigroups have positive upper density?

Let $\mathbb{G}$ be a discrete amenable semigroup, and $\left\{ F_{n}\right\} $ a Folner sequence. For $S\subset \mathbb{G}$ define the upper density as $D^{\ast }(S)=\limsup_{n\rightarrow \infty ...
6
votes
1answer
127 views

Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?

Let $G$ be a locally compact (second countable) group and let $$ G_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}. $$ This is the kernel of the ...
6
votes
3answers
314 views

A Hausdorff abelian group with no character?

Pontryagin Duality for locally compact abliean groups gives plenty of continuous (unitary) characters $\chi : A \to \mathbb{R} / \mathbb{Z}$, but if we do not assume local compactness, can anything be ...
7
votes
0answers
188 views

Which topological spaces are coset spaces of locally compact groups?

What is a topological characterization of the class of spaces that have the form $G/H$ for a locally compact, Hausdorff group $G$ and a closed subgroup $H$ ? Such a space $X=G/H$ necessarily ...
-4
votes
1answer
210 views

Is the product of closed subgroups in a locally compact group locally compact? [closed]

Let $A$ and $H$ be closed subgroups of a $\sigma$-compact locally compact group $G$. Assume further that $A$ is abelian. Is the group $AH$ locally compact subgroup in the subspace topology?
1
vote
1answer
254 views

Mean value theorems for the Haar integral?

Let $G$ be a compact topological group (feel free to add hypotheses if necessary). Is there any mean value theorem for its (normalized to 1) Haar integral? In general, are there mean value theorems ...
1
vote
0answers
191 views

does s.e.s 0->A->B->C->0 of profinite groups imply C=B/A and A<B topologically?

Assume $A, B, C$ are profinite groups and $0\to A\to B\to C\to 0$ is an exact sequence of continuous maps. Which of the following assertions follows?: (i) the subspace-topology induced on $A$ via ...
1
vote
0answers
134 views

Intersection of cocompact closed normal subgroups

Let $G$ be a locally compact Hausdorff topological group. Definition A closed normal subgroup $H \unlhd G$ is called cocompact if $G/H$ is compact with respect to the quotient topology. Note ...
11
votes
2answers
353 views

Start with a topological group, take the meet of the two uniformities, and take the topology. Is the result again a topological group? [xpost from math.SE]

And what else can be said, if so? (Original math.SE post) In more detail: Say $(G,\mathscr{T})$ is a topological group. It has a left uniformity $\mathscr{L}$ and a right uniformity $\mathscr{R}$. ...
4
votes
0answers
87 views

A dynamical property of automorphisms of a locally compact group

Let $G$ be a Hausdorff locally compact group and let $\alpha$ be an automorphism of $G$. Say $\alpha$ is (forwards) topologically recurrent if for all $g \in G$ and all neighbourhoods $O$ of $g$, the ...
7
votes
0answers
218 views

Strange normal subgroups of profinite groups

I am looking for an example of the following situation: $G$ is an infinite profinite group, with a dense normal subgroup $N$. However $N$ does not contain any non-trivial closed normal subgroup of ...
5
votes
5answers
562 views

Commutator of algebraic subgroups is connected

Let $G$ be an algebraic group over an algebraically closed field. If $H$ and $K$ are closed subgroups and one of them is connected, then their commutator $[H,K]$ is also connected. Is ...
6
votes
3answers
892 views

$\pi_1$ Sequence of Topological Groups

Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
1
vote
1answer
351 views

Finiteness theorems for profinite groups

Let $G$ be a profinite group which fits in the following short exact sequence: $$ 1\rightarrow N\rightarrow G \rightarrow K\rightarrow 1 $$ Assume that $N$ is a pro-$p$ group and that $K$ is ...
8
votes
0answers
251 views

'Infinitesimal' elements of a topological group

Let $G$ be a topological group, and let $M$ be the intersection of all conjugacy-invariant neighbourhoods of the identity in $G$ (in other words, the set of elements that can be taken arbitarily close ...
5
votes
0answers
198 views

Examples of a non-Hopfian phenomenon in group theory

I am interested in examples of the following property, where $G$ is a non-discrete locally compact topological group: (*) The open normal subgroups of $G$ have trivial intersection, but $G$ has an ...
6
votes
2answers
905 views

Two Definitions of “Character” of topological groups

When I first met the concept of "characters" of topological groups in Pontryagin's book "Topological groups", it was defined as follows: Let $G$ be a topological group. A character of $G$ is a ...
12
votes
1answer
903 views

(Closures of sets of) operations in topological groups.

Let $G$ be a topological group. For each $n \in \mathbb{Z}$, consider the continuous functions $f_{n} \colon G \to G : x \mapsto x^{n}$, and set $F := \{f_{n} \mid n \in \mathbb{Z}\}$. Is there a ...
0
votes
1answer
517 views

No non-trivial homomorphism to a group

Here is a question I posted some months ago in Math.SE, and t.b. mentioned to the following question by Florent MARTIN which is somehow related to my question; Let $G$ be a compact Hausdorff ...
20
votes
2answers
2k views

morphism from a compact group to Z ?

I wonder if it there exists a topological compact group $G$ (by compact, I mean Hausdorff and quasi-compact) and a non-zero group morphism $\phi : G \to \mathbb{Z}$ (without assuming any topological ...
0
votes
0answers
254 views

classifying continuous functions on the $ax+b$ subgroup of $GL(2,\mathbb R)$

Let $G$ be 'ax+b' topological group i.e subgroup of $GL(2,\mathbb R)$ containing $2\times 2 $ matrices of type $\{\left(\begin{matrix} a&b\\\0&1\end{matrix}\right): a\neq 0\; a,b\in \mathbb ...
3
votes
2answers
258 views

When does a LCA group not contain a (closed) infinite cyclic subgroup?

If $G$ is an LCA (locally compact abelian) group, is there any 'nice' sufficient (or preferably necessary and sufficient) criteria for when $G$ does not contain a closed (and hence discrete in the ...
6
votes
4answers
691 views

Criteria for topologically finitely generated profinite groups

Q1: Do we have a criterion which allows us to say when is a profinite group $G$ topologically finitely generated? For example, if $G$ is topologically finitely generated then, for a fixed integer ...
20
votes
0answers
860 views

Do all possible trees arise as orbit trees of some permutation groups?

I.Motivation from descriptive set theory (Contains some quotes from Maciej Malicki's paper.) The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...
10
votes
1answer
413 views

Epimorphisms have dense range in TopHausGrp?

Consider the category of Topological Groups with continuous homomorphisms. Then a continuous homomorphism $f:G\rightarrow H$ with dense range is an epimorphism. Is the converse true? If not, what ...
11
votes
1answer
689 views

Non-isomorphic two-transitive permutation groups with isomorphic point stabilizers

The permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}_8$ ...
7
votes
2answers
1k views

Locally compact abelian groups

First, some preliminaries: Define an "LCA group" to be a locally compact Hausdorff abelian topological group. Define "smooth manifold" in a way that requires Hausdorffness, but not connectedness or ...
2
votes
2answers
969 views

Irreducible unitary representations of locally compact groups

Let $G$ be a locally compact group and let $\mu$ be a left Haar measure. We know that $\mu$ is unique up to a scalar in $\mathbf{R}_{>0}$. I don't know so much about unitary representations of ...
7
votes
1answer
474 views

index of a closed subgroup of a profinite group

In the book "profinite groups, arithmetic, and geometry" of Shatz, the index $(G:H)$ of a closed subgroup $H$ of a profinite group $G$ is defined to be the supernatural number $lcm\big((G/U):(H/(H\cap ...
5
votes
4answers
502 views

orbits in locally compact group

As everyone knows if $x\in S^1$, then the set $\{ x^n \}$ is either finite or dense. Under which condition is true for any other locally compact group, i.e if $G$ is a locally compact group, and $x\in ...
36
votes
6answers
2k views

Does homeomorphic and isomorphic always imply homeomorphically isomorphic?

Let $(G,\cdot,T)$ and $(H,\star,S)$ be topological groups such that $(G,T)$ is homeomorphic to $(H,S)$ and $(G,\cdot)$ is isomorphic to $(H,\star)$. Does it follow that $(G,\cdot,T)$ and ...
-1
votes
2answers
638 views

Is there a group homeomorphic to but not homeomorphically isomorphic to the circle group?

Let $Circ$ be the topological group $(\{z\in \mathbb{C} : \overline{z}\cdot z = 1\},\cdot , \{U\in 2^{\{z\in \mathbb{C} : \; \overline{z}\cdot z \, = \, 1\}} : \{z\in \mathbb{C} : \overline{z}\cdot z ...
4
votes
1answer
205 views

existence of charaterization of amenable groups by complementation?

Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$. Do you know a charaterization of discrete amenable ...
8
votes
3answers
513 views

Congruence Subgroups as Open Subgroups of the Modular Group Under the Right Topology

It occurred to me that a subgroup of the modular group $\Gamma$ is a congruence subgroup iff it contains a subgroup of the form $\Gamma(N)$, while a subgroup of a general topological group is open iff ...