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4
votes
1answer
196 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
204 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 ...
3
votes
2answers
196 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
111 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
232 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 ...
4
votes
1answer
246 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
103 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
308 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
100 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
96 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 ...
5
votes
1answer
203 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
158 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
106 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
75 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
130 views

Is every first countable profinite group, second countable?

Is every first countable profinite group actually second countable?
14
votes
3answers
461 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
128 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
482 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
257 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
83 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
129 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
92 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
188 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
279 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
715 views

Profinite groups as étale fundamental groups

Does every profinite group arise as the étale fundamental group of a connected scheme? Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme? ...
2
votes
0answers
73 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
307 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
208 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
266 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
335 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
216 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 ...
3
votes
1answer
197 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
97 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
335 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
358 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 ...
10
votes
0answers
314 views

Higher-dimensional algebraic subgroups of the proalgebraic Nottingham group?

Let $R$ be a commutative ring, and, for $n\ge0$, ${\mathcal{A}}_n={\mathcal{A}}_n(R)$ the group of series $u(x)=\sum_0^\infty a_jx^{j+1}\in R[[x]]$ for which $a_0\in R^\times$ and $u(x)\equiv ...
5
votes
2answers
788 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 ...
8
votes
4answers
1k views

Topological examples of profinite groups

I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first ...
4
votes
1answer
203 views

How to determine free generators of a closed subgroup of a free pro-$p$-group ?

If $F$ is a free discrete group, then any subgroup $H$ of $F$ is free: this is the well-known theorem of Nielsen-Schreier. Moreover, there is a well-known algorithm, the Nielsen-Schreier method that ...
6
votes
2answers
371 views

A metabelian quotient of a free group

I don't know much about free groups (excepted the very basics), and the following question may be trivial, although it isn't to me. Let $F$ be a free group with $n$ generators $x_1,\dots,x_n$. ...
20
votes
4answers
1k views

A profinite group which is not its own profinite completion?

Is there a profinite group $G$ which is not its own profinite completion? Surely not, I thought. But upon looking into it, I found that there is a special name given to a $G$ which is its own ...
1
vote
1answer
228 views

a question on continuity of $G$-module for a profinite group $G$

I have seen the following statment somewhere, for example in Appendix B2 on Silverman's book "The Arithmetic of Elliptic Curves" : Let $M$ be an abelian group with discrete topology and $G$ be a ...
4
votes
2answers
474 views

Is a profinite group with a finite number of simple quotients and Jordan-Hölder factors finitely generated?

Assume $G$ is a profinite group such that the Jordan-Hölder factors appearing in the finite quotients vary in a finite number of isomorphism classes of simple groups. Assume also $G$ to have a finite ...
15
votes
3answers
817 views

Homomorphism from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$

I expect this question has a very simple answer. We all know from primary school that there are no non-trivial continuous homomorphisms from $\hat{\mathbb{Z}}$ to $\mathbb{Z}$. What if we forget ...
12
votes
3answers
1k views

Finitely generated Galois groups

It is well-known that for a given natural number $n$ there is only finite number of extensions of $\mathbb Q_p$ of degree $n$. This result appears in many introductory books on algebraic number ...
7
votes
2answers
557 views

Centralizers of elements in free profinite groups

I believe, although I can't say that I've given a rigorous proof, that for a free group $F_r$, and an element of it $a$, $C_{F_r}(\langle a \rangle)=$ the group generated by the elements $b \in F_r$ ...
10
votes
3answers
692 views

Profinite completion of a semidirect product

If we have two finitely generated residually finite groups $G$ and $H$, is there are relation between the profinite completions $\hat{G},\hat{H}$ and the profinite completion of a semidirect product ...
0
votes
1answer
233 views

how large can this pro-p quotient be?

Let $p$ and $\ell$ be distinct rational primes. Note that the unit group of the finite field $\mathbb{F}_\ell$ is of order $\ell-1$, hence there is the probability of finding a $p$-quotient from ...
19
votes
4answers
2k views

What is the virtue of profinite groups as mathematical objects?

In my own research I use profinite groups quite frequently (for Galois groups and etale fundamental groups). However my use of them amounts to book-keeping: I only care about finite levels (finite ...
25
votes
3answers
2k views

Why are profinite topologies important?

I hope this is not too vague of a question. Stone duality implies that the category Pro(FinSet) is equivalent to the category of Stone spaces (compact, Hausdorff, totally disconnected, topological ...