# Questions tagged [profinite-groups]

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• 521
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### If $F$ is a prosoluble subgroup of a free profinite product $\amalg G_i$ and $F \cap G_i^g$ is pro-$p$, is also $F$ pro-$p$?

There is a 1995 paper (Manusc. Math., DOI link) of Florian Pop where he proves the following: Theorem. Let $G$ be the free product of profinite groups $G_i$ and $F$ a closed prosoluble subgroup of $G$...
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### When is a group the same as its profinite completion

I'm working with the inner automorphism group of profinite quandles. A question I have yet to resolve is whether or not the inner automorphism group of a profinite quandle is necessarily profinite, or ...
• 253
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### Stone-topological/profinite equivalence for quandles

A quandle $(Q,\triangleleft,\triangleleft^{-1})$ is a set $Q$ with two binary operations $\triangleleft,\triangleleft^{-1}:Q\times Q\to Q$ such that the following hold for all $x,y,z\in Q$: (Q1) ...
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### Topological generators for the Sylow pro-$p$ subgroup of $\mathrm{SL}_2(\mathbf{Z}_p)$

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $G_2(\mathbf{Z}_p):=\begin{pmatrix} 1+p\mathbf{Z}_{p} & \mathbf{Z}_{p}\\ p\mathbf{Z}_{p} & 1+p \mathbf{Z}_{p} \end{pmatrix}$. ...
1 vote
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### A closed subgroup of $p$-adic analytic group having same dimension is open?

Let $G$ be $p$-adic analytic pro-$p$ group and $H$ a closed subgroup of $G$. Suppose that $G$ and $H$ have the same dimension as $p$-adic analytic groups. Question: Is it true that $H$ is an open ...
169 views

### Open conjugacy classes in a second countable profinite group

Let $G$ be a second countable profinite group, $g\in G$ and $g^G:=\{hgh^{-1}~|~h\in G\}$ the conjugacy class of $g$ in $G$. Theorem 3.2 in Wesolek's Conjugacy class conditions in locally compact ...
• 521
1 vote
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### Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups

While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
• 1,043
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### Convergent condition in the definition of free profinite groups

I have a (perhaps simple) question about free pro-$C$ constructions. Definition. Let $X$ be a set. The free pro-$C$ group over $X$ is a pro-$C$ group $F$ together a $1$-convergent map $f: X \to F$ ...
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• 289
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### “Sheaf cohomology” of Galois groups

Apparently the Galois group $G$ of a Galois extension $E/F$ can be viewed as an “étale sheaf” on the set $X$ of intermediate Galois extensions equipped with an appropriate Grothendieck topology (see ...
• 301
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### Is insoluble $p$-adic analytic just-infinite pro-$p$ group torsion-free?

Recall that an infinite pro-$p$ group $G$ is called just-infinite if all non-trivial closed normal subgroup of $G$ have finite index. Question: Let $G$ be an insoluble $p$-adic analytic just-infinite ...
• 521
1 vote
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### Quotients of pro-$p$ groups linear over a complete Noetherian local ring

Let $R$ be complete Noetherian local ring with finite residue field $\mathbb{F}$ of characteristic $p$. We say that a pro-$p$ group $G$ is linear over $R$ if it is isomorphic to a closed subgroup of ...
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### For locally profinite groups $H\lhd G$, is there a spectral sequence $\newcommand\@[2]{{\rm Ext}_#1^{#2}(\pi_1,\pi_2)}H^p(G/H,\@Hq)\implies\@G{p+q}$?

Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi_1$ and $\pi_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows? ...
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### Endo reversible words

Let $w$ be a word in free group $F$ on finitely many generators. We will look at $w$ as word map on groups. It is clear that there exists an endomorphism $\phi$ of $F$ such that $\phi(w) = w^{-1}$ if ...
• 335
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### Subgroup growth of direct product

I have started reading about subgroup growth and, to my surprise, I haven't found a reference to whether direct products preserve subgroup growth. Recall that, given a finitely generated group $G$, ...
• 541
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### Structure of a profinite group as a condensed set with an action of an open subgroup

Let $G$ be a profinite group and $H$ be an open subgroup. As a continuous $H$-topological space, we have $G=\coprod_{G/H} H$. Does this also hold as condensed sets, i.e. do we have an identification ...
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### An infinite profinite group such that any $\overline{\mathbb{F}_{p}((t))}$-adic representation has finite image
This question is a sequel to An infinite profinite group such that any $p$-adic representation has finite image . Fix a prime $p$. We call an infinite profinite group $G$ a Boston group (with ...