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2
votes
1answer
236 views

Is every closed subgroup of $\text{GL}_n(K[[x]])$ finitely generated?

Let $n \in \mathbb{N}$, $K$ a finite field. Denote by $K[[x]]$ the (profinite) ring of formal power series over $K$. Note that $\text{GL}_n(K[[x]])$ is a profinite group. Is every closed subgroup of ...
3
votes
1answer
134 views

Faithful representations of free pro-p groups

Let $p$ be a prime number, $m,n \in \mathbb{N}$, $F = F(p,m)$ be the free pro-$p$ group on $m$ generators. For which $(m,n)$ there is a continuous faithful representation (embedding) $\rho : F ...
2
votes
1answer
224 views

Can a closure make the index finite?

Let $F$ be a free finitely generated group, $H \leq F$ of infinite index. Let $c : F \rightarrow \hat{F}$ be the embedding in the profinite completion. Denote by $\tilde{F}, \tilde{H}$ the closure of ...
2
votes
1answer
178 views

Making a profinite group free

Let $F$ be a free profinite group, $G$ a profinite group. Suppose that the free profinite product $F \amalg G$ is a free profinite group. Must $G$ be a free profinite group? For abstract groups the ...
6
votes
2answers
215 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 ...
2
votes
1answer
169 views

Bases of free groups

Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ ...
4
votes
1answer
140 views

Generators of Sylow subgroups

Is there a function $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for each finite supersolvable group $G$, and a Sylow subgroup $S \leq G$ we have $d(S) \leq f(d(G))$? Here $d(H)$ denotes the ...
0
votes
0answers
90 views

Actions and representations of profinite groups

Let $p$ be a prime number, and denote by $\mathbb{Z}_p$ the additive profinite group of p-adic integers. Let $G$ be a finitely generated profinite group of order coprime to $p$, and $V = ...
3
votes
0answers
175 views

Profinite groups, completions, and Schreier's formula

Let $G$ be a finitely generated profinite group, and $H \leq_o G$. We say that $H$ satisfies Schreier's formula in $G$ if $d(H) - 1 = (d(G)-1)[G:H]$. We say that $G$ satisfies Schreier's formula if ...
4
votes
2answers
291 views

Profinite completions

I call a profinite group $G$ Noetherian, if evrey ascending chain of closed subgroups is eventually stable. A standart argument shows that every closed subgroup of a Noetherian profinite group is ...
4
votes
0answers
174 views

Schreier's formula and descending chains

For a group $G$ we denote by $d(G)$ the cardinality of a smallest set of generators. A finitely generated group $G$ is said to satisfy Schreier's formula if for every subgroup $H \subseteq G$ of ...
2
votes
0answers
84 views

Coprime automorphisms of finitely generated pro-$p$ groups

Let $P$ be a finitely generated pro-$p$ group and let $G$ be a semidirect product $P \rtimes A$, where $A$ is a finite group of order coprime to $p$ that acts faithfully on $P$. Then one can show ...
2
votes
0answers
78 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 $n$?
5
votes
1answer
180 views

Is any finitely generated nilpotent pro-$p$ group necessarily the pro-$p$ completion of some finitely generated nilpotent group?

While thinking about this question, I was led to the following question: My question: Let $G$ be a topologically finitely generated pro-$p$ nilpotent group. Does there exist a finitely generated ...
4
votes
3answers
519 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 ...
4
votes
1answer
246 views

Normal Subgroup Growth

Let $F$ be a free group on $d$ generators. Denote by $F_{k}$ the $k$-th term in $F$'s derived series. Put $G = F/F_k$. What is the normal subgroup growth of $G$? Explicitly, for each natural number ...
4
votes
2answers
244 views

Schreier's index formula

A finitely generated group G is said to satisfy Schreier's index formula if for every subgroup H of index k in G we have: d(H) - 1 = k(d(G) - 1). For example, a finitely generated free group satisfies ...
3
votes
1answer
321 views

Relations between the cohomology of discrete groups and of profinite groups

Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for ...
5
votes
1answer
247 views

Open subgroups of the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$

Let $G$ be the etale fundamental group of $P^1_\mathbb Q\setminus\{0,\infty\}$. Then $G$ is isomorphic to a semidirect product of $\widehat {\mathbb Z}(1)$ by $ Gal_\mathbb Q$. Is it true that ...
5
votes
1answer
259 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 ...
3
votes
0answers
115 views

Computing the pro-solvable closure of a finitely generated subgroup of a free group

The pro-solvable topology on a group $G$ is the unique group topology such that the set of normal subgroups $N\lhd G$ with $G/N$ a finite solvable group is a fundamental system of neighborhoods of the ...
4
votes
2answers
333 views

Irreducible representations of compact groups

Let G be a compact group (or even profinite - Galois group). Let $V$ be a vector space over the field ${\mathbb F}_p$ with $p$ elements, $p$ a finite prime, such that $V$ is a contable product of ...
2
votes
0answers
107 views

Hall's paper on the profinite groups and Andre Weils “voisinage” notion

I am reading through a classical paper A Topology for Free Groups and Related Groups by Marshall Hall Jr. in which profinite groups are defined for the first time. There he defines on p. 129: ...
1
vote
0answers
103 views

Local Profinite Ring

I haven't received any substantial responses to a similar question on math.stackexchange, so let me try here. Let $R$ be a profinite ring (that is a projective limit of finite rings). Assume ...
6
votes
1answer
230 views

“Concretely” writing down elements in a free profinite group

Let $r$ be a natural number. The elements of the free group $F_r$ on $r$ generators have a nice concrete description as "words" in the $r$ generators (and their inverses). I'd like to know if there is ...
4
votes
1answer
168 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 ...
4
votes
1answer
110 views

Layman question: A dense subgroup with completion not isomorphic to the big (pro-p) group?

This is unlikely a research level question... one that would be answered in a blink of an eye, rather...it is an (early) exercise from the book "Analytic Pro-p groups". But since no reply was received ...
4
votes
1answer
84 views

Is every countably generated profinite group countably based?

In a profinite group: Does the existence of a countable generating (topologically) set imply the existence of a countable basis for the topology.
3
votes
2answers
150 views

Is every first countable profinite group, second countable?

Is every first countable profinite group actually second countable?
14
votes
3answers
469 views

An algebraic approach to the thermodynamic limit $N\rightarrow\infty$?

In physics one studies quite often the thermodynamic limit or what we call the $N\rightarrow \infty$ behavior of a system of $N\rightarrow\infty$ particles. This is of particular relevance in the ...
0
votes
1answer
135 views

Can finite index be seen at the level of profinite completion

Let $G$ be a group, and $H$ a subgroup of $G$. Is it possible to "see" from the profinite completions of $H$ and $G$ that $H$ has finite index in $G$? Naively, does $H$ have finite index in $G$ iff ...
9
votes
3answers
500 views

History of profinite groups, when was it first mentioned? What was the original definition?

Searching left me hanging. One of my professors told me the definition using the topological properties was the first one but I cannot find any resources. Is that true? If not, how was it originally ...
6
votes
1answer
273 views

Are finite index subgroups of inertia closed?

Let $K$ be a finite extension of the $p$-adic numbers. $G_K$ be its absolute Galois group and $I_K$ the inertia subgroup. Are finite index subgroups of $I_K$ closed in its profinite topology? By a ...
1
vote
1answer
85 views

Quotients of Free pro-p groups

Let $P_n$ denote the pro-$p$ completion of $F_n$ the free group of rank $n$. Given a (abstract) group homomorphism $$ \phi:P_n\rightarrow G $$ where $G$ is a discrete group. Is $\phi$ continuous? ...
0
votes
1answer
146 views

Strongly Complete Profinite Groups.

I've been reading about profinite groups and have encountered the notion of strong completeness. I.e. that a profinite group $G$ is strongly complete if it is isomorphic to it's profinite completion ...
3
votes
0answers
94 views

Centralizers in free products of $p$-groups

If $G_1,\dots,G_n$ are discrete groups, and $G=G_1 \ast G_2 \ast \dots \ast G_n$ is their free product, then for $g \in G_i$ sean as an element of $G$, it is clear the centralizer of $g$ in $G$ is ...
1
vote
0answers
190 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 ...
2
votes
2answers
287 views

Are extensions of profinite groups profinite?

Assume $X$, $E$ and $G$ are topological groups and $1\to X\to E\to G\to 1$ a short exact sequence of continuous group homomorphisms. Under which of these conditions is $E$ a profinite group? (i) $G$ ...
21
votes
2answers
733 views

Profinite groups as étale fundamental groups

Does every profinite group arise as the ├ętale fundamental group of a connected scheme? Equivalently, does every Galois category arise as the category of finite ├ętale covers of a connected scheme? ...
2
votes
0answers
76 views

Infinitely generated powerful pro-$p$ groups

A pro-$p$ group of finite subgroup rank has an open subgroup $P$ that is uniformly powerful, meaning that $[P,P]$ is contained in the group generated by $2p$-th powers in $P$, and raising elements to ...
5
votes
0answers
337 views

Examples of uncountable abelian $p$-groups

Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable? By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, and totally ...
7
votes
0answers
216 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 ...
2
votes
2answers
288 views

When is the semidirect product of profinite groups a profinite group?

Following the discussion I have with Yves Cornulier in the following question Finiteness theorems for profinite groups, I would like to ask the following: Suppose $K$ and $N$ are two profinite groups ...
1
vote
1answer
348 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 ...
1
vote
1answer
230 views

Neighborhood basis of the identity in a locally profinite group

Consider a locally profinite group $G$, i.e. a locally compact, totally disconnected topological group. Suppose it admits an open maximal compact subgroup named $K$. It is known that $G$ admits as a ...
4
votes
1answer
222 views

Haar measure for profinite groups (reference needed)

I was wondering if anybody knows a good reference book or exposition for Haar measures over profinite groups (with some concrete examples and computations)?
4
votes
1answer
106 views

Open subgroups of free pro-C groups

This question is related to this mathoverflow question that I've asked recently. The question rose while I prepared my lectures on Profinite Groups in an advance course in Tel Aviv University. Let ...
6
votes
1answer
351 views

Open subgroups of free profinite groups

The following questions popped out while I was preparing a course on profinite groups. Closed subgroups of free profinite groups are not necessarily profinite free (e.g. the p-sylow subgroups, or ...
8
votes
1answer
361 views

Homomorphic images of a Cartesian product of finite groups

What can be said about the class of groups which can be represented as a homomorphic image of an (infinite) Cartesian product (=unrestricted direct product) of finite groups? What would be simple ...