The profinite-groups tag has no wiki summary.

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### Is there a left orderable profinite group?

Is there a profinite group $G$ with a binary transitive relation $<$ such that for any different $x,y \in G$ either $x < y$ or $y < x$ and such that for any $x,y,z \in G$ we have that $x < ...

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### Is there a nontrivial profinite word which is trivial in any group with at most d generators?

Let $F$ be a free profinite group of rank $\aleph_0$, and let $d \in \mathbb{N}$.
Let $N_d \lhd_c F$ be the intersection of all open normal subgroups $L \lhd_o F$ for which $F/L$ can be generated by ...

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### Exhausting a free pro-p group

Recall that for a profinite group $G$ we define the subgroup rank to be $$\sup \{d(H): H \leq_c G\}$$ where $d(H)$ stands for the minimal cardinality of a set of topological generators of $H$.
Let ...

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### A Karrass-Solitar or Ivanov-Schupp for profinite groups

Let $F$ be a nonabelian free profinite group, $H \leq_c F$ finitely generated with $[F:H] = \infty$. Must there be some $\{1\} \neq N \lhd_c F$ such that $N \cap H = \{1\}$?

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### Thin profinite groups - nonabelian analogues of p-adic integers

Let $p$ be a prime number, $S = C_p$ a cyclic group of order $p$, $G = \mathbb{Z}_p$ the profinite additive group of $p$-adic integers. It is well known that all the closed nontrivial subgroups of $G$ ...

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### discrete group cohomology vs continuous group cohomology for profinite groups

Let $G$ be a profinite group and $M$ be a finite $G$-module. I can compute the cohomology of $G$ with coefficients in $M$ either as a topological group or as a discrete group. There is an obvious map ...

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### An epimorphism into a profinite group

Let $p$ be an odd prime number, $G$ a finitely generated nonabelian profinite group, $L \lhd_o G$ a pro-$p$ group with $[G : L] = 2$. Suppose that there is a continuous surjection from $G$ onto a free ...

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### Bases for free pro-p groups

Let $p$ be a prime number, $F$ a free nonabelian finitely generated pro-$p$ group, $L \lhd_o F$ and $Y$ a basis for $L$ with $y \in Y$. Is there a basis $X$ for $F$ such that $y$ is in the abstract ...

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### Dense free subgroups

Let $F$ be a free pro-$p$ group (for a prime number $p$) on a finite set $X$, $\Phi$ the abstract subgroup generated by $X$, $\{1\} \neq N \lhd_c F$. Is it possible that $\Phi \cap N = \{1\}$?

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### Lower central series in a free pro-p group

Let $F$ be a nonabelian finitely generated free pro-$p$ group, $H \leq_c F$ of infinite index. Denote by $\{F_n\}_{n \in \mathbb{N}}$ the lower central series of $F$, and set $r_n = [F : F_nH]$.
Is ...

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### A bound on the size of the center

Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$

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### Finite Cohomology and free groups

Let $F$ be a finitely generated nonabelian free profinite group, $p$ a prime number, $L \lhd_o F$ with $[F : L]$ coprime to $p$, $N \lhd_c^\infty F$ contained in $L$ with $L/N$ pro-$p$, and $N \leq H ...

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### Accessible subgroups of free groups

Let $F$ be a nonabelian free finitely generated group, and $F = G_0 \rhd G_1 \rhd G_2 \dots$ a strictly descending subnormal chain of subgroups ($G_n \lhd G_{n-1}$ for each $n \in \mathbb{N}$) each ...

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### Canonical presentation of pro-modules over pro-rings

Let $A = (\dotsc \twoheadrightarrow A_2 \twoheadrightarrow A_1 \twoheadrightarrow A_0)$ be a (commutative) pro-ring with surjective transition maps. Consider the category $\mathcal{M} := \varprojlim_i ...

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### When is the profinite completion a pro-$p$ group?

My research area is mainly pro-$p$ groups and profinite groups. However, in the last few year I became also interested in discrete groups. Therefore, it seems to me a natural problem to look for ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $H$ ...

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### 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 ...

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### 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 = ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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 $n$?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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:
...

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### 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
...

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### “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 ...

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### 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 ...

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### 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 ...

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### 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.

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### Is every first countable profinite group, second countable?

Is every first countable profinite group actually second countable?

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?
...