# Questions tagged [profinite-groups]

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### Non-abelian group cohomology, additional information

Let $G$ be a (profinite) group, and let $M$ be a non-abelian $G$-module. We know how to construct reasonably $H^0(G,M)$ and $H^1(G,M)$ and it turns out that $H^1(G,M)$ is just a pointed set and not ...
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### Countability of conjugacy classes in profinite groups

In the MOF question  it was asked if $G$ is a second-countable profinite group with uncountably many subgroups, does it follow that it has uncountably many closed subgroups modulo conjugacy? A ...
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### Colimits in cohomology of profinite arithmetic groups

Let $G\subset \operatorname{GL}_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap \operatorname{GL}_n(\mathbb{Z})$ be an arithmeric subgroup without torsion. Using a result ...
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### Rational cohomology of p-adic general linear groups

I wanted to compute the cohomology ring $H^*(GL_n(\mathbb{Z}_p); \mathbb{Q}_p)$ (with $p$ fixed prime as usual). I found some incomplete notes stating that the computation should go as follows. First ...
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### Infinite pro-$p$ group of finite solvable length and finite coclass

I was reading about infinite pro-$p$ groups of finite coclass from the book "The Structure of Groups of Prime Power Order" by Leedham-Green and McKay. I asked this question in math....
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Let $\mathit{Profinite}_{\mathrm{Ab}}$ be the category of profinite abelian groups, and let $\mathit{Profinite}_{\mathrm{Set}}$ be the category of profinite sets. Does the forgetful functor $$\mathit{... 4 votes 1 answer 268 views ### Measure of subsets of profinite groups Let G be an infinite profinite group, so$$G=\lim_{\longleftarrow}G/N where $N$ runs through the open normal subgroups. I have two questions: Is $G$ of Haar measure zero in the compact group $\... 3 votes 1 answer 504 views ### Continuous function defined by measurable sets Is the following slightly generalization of Corollary 20.17 in Hewitt and Ross Book (page 296) correct? Let$A$be a subset of a profinite group$G$( compact, Hausdorff, totally disconnected ... 5 votes 2 answers 408 views ### Mackey theory in the setting of locally profinite groups$\DeclareMathOperator\Hom{Hom}$Let$R$be a commutative ring (not necessarily unital). Let$G$be a finite group, and let$H_1, H_2$be subgroups of$G$. Recall the following standard result [1, Thm. ... 12 votes 1 answer 682 views ### Does a (nice) centerless group always have a centerless profinite completion? This is an extension of a question I asked here on Math.SE Assume that I have a finitely generated residually finite centerless group$G$. Is it true that the profinite completion$\hat{G}$also has ... 1 vote 2 answers 278 views ### Meaning of epimorphism from full Galois group to some group My problem has two parts: let$\;G:=\operatorname{Gal}(\overline{\Bbb Q}/\Bbb Q)\;$be the full Galois group of the rationals and$\;K\;$be some finite group, then: (1) Does having an epimorphism (... 5 votes 0 answers 194 views ### Applications of one of Serre's Theorems This theorem is due to Serre: Let$G$be a profinite group,$p$prime. Assume that$G$has no element of order$p$and let$H \leq G$be an open subgroup. Then$cd_p(G) = cd_p(H)$. Where$cd_p(...
Let $\widehat F(k)$ be the free profinite group on $k$ generators and let $p$ be a prime. Then there is a canonical projection $\pi\colon \widehat F(k)\to \widehat F_p(k)$ where $\widehat F_p(k)$ is ...