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Let $p$ be a prime number, and let $G$ be a pro-$p$ group with finite generator rank $d(G)$ and finite relation rank $r(G)$. If $G$ satisfies the Golod--Shafarevich condition $r(G) < d(G)^2/4$ and ...
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$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $p$ be a prime number, $G$ a pro-$p$ group, and $m$ a positive integer. We say that $H(G,m)$ holds if there is a function of $m$ that is an ...
stupid boy's user avatar
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Let $G$ be a pro-$p$ group. Consider the graded algebras $$H^*(G,\mathbb{Z}/p^n\mathbb{Z}).$$ Does the structure of these graded algebras determine $G$ up to isomorphism? Feel free to add additional ...
Snacc's user avatar
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Let $p$ be an odd prime and $\mathbb{Z}_{p}$ be the ring of $p$-adic integers. Let $\mathbb{F}_{p}$ be the finite field of order $p$ and let $\mathbb{F}_{p}[[T]]$ be the ring of formal power series ...
stupid boy's user avatar
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Let $K$ be a non-archimedean local field of characteristic $p>0$ and $\mathcal{O}_{K}$ its valuation ring with maximal ideal $\mathfrak{m}$. Let $\mathbf{G}$ be a connected, simply connected $K$-...
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We know that for a pro-$p$ group $G$, it is topologically finitely generated if and only if the first cohomology group $H^1(G,Z/pZ) $ is finite. I am wondering if a similar characterization exists for ...
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Let $F$ be the free profinite group of rank $r$. Let $\tilde{F}$ be the free profinite group of rank $r + 1$, and $C$ be a subgroup of $\tilde{F}$ which is, of course non-canonically, isomorphic to $\...
Alice's user avatar
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Let $\mathbb{F}_{p}[[T]]$ denote the ring of formal power series over the finite field $\mathbb{F}_{p}$ of order $p$. Let ${\rm SL}_{2}^{1}(\mathbb{F}_{p}[[T]])$ denote the first congruence subgroup ...
stupid boy's user avatar
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2 answers
388 views

$\DeclareMathOperator\Cor{Cor}\DeclareMathOperator\Res{Res}$Let $G$ be a profinite group, $H$ an open subgroup (necessarily of finite index). Let $X$ and $Y$ be $G$-modules, and let $f : X \to Y$ be ...
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I am interested in the role of profinite duality groups in Galois theory. In this context, I would like to understand the following result of Brady and Meier: A right-angled Artin group $A_{\Gamma}$ ...
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Let $F$ be a totally real field of even degree and let $f$ be a Hilbert cusp form of level $\mathfrak{n}$ that is an eigenform for Hecke operators. Let $p$ be an odd prime number and $\mathfrak{p}$ a ...
Zakariae.B's user avatar
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Let $p$ be prime and let $\mathbb{Z}_p$ be the ring of $p$-adic integers. Let $G$ be a (topologically) finitely presented pro-$p$ group, and let $F$ be a closed normal subgroup of $G$. Suppose that $F$...
stupid boy's user avatar
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Let $G$ be a profinite group on finitely many topological generators, say $r$, subject to one relation, that they all multiply to the identity. Let $M\simeq (\mathbb{Z}/\ell \mathbb{Z})^{n}$ be a ...
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This question asks if a Sylow subgroup of a finitely generated profinite group $G$ is also finitely generated. The answer is negative. The counterexample to the claim involves infinitely many primes. ...
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Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...
David Corwin's user avatar
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Let $f$ be a continuous endomorphism of a compact Hausdorff totally disconnected topological group $G$ and let $H$ be a closed normal subgroup of G such that $f(H)\subseteq H$ and with $\mu(H)=0$ ...
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$\DeclareMathOperator\ind{ind}\DeclareMathOperator\res{res}$ My question: what is known or expected to be true about the existence of a Mackey decomposition formula in the case of closed subgroups of ...
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Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
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It is shown in [1, Section 1] by C.W. Onneweer that every infinite compact, metrizable, zero-dimensional commutative group is a Vilenkin group. My question is does this implication also hold if we ...
user avatar
9 votes
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Let $p$ be a prime. Let $w$ be an element of a free pro-$p$ group $F_r$ of finite rank $r\geq 2$. Then we say that a pro-$p$ group $G$ satisfies the pro-$p$ identity $w$ if for every homomorphism $ f:...
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It's proved in Oltikar and Ribes - On prosupersolvable groups that any Sylow pro-$p$ subgroup of a topologically finitely generated prosupersolvable group is also topologically finitely generated. It ...
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Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings. Then if $f_{n,n+1}: \Bbb{Z}/p_{...
Luna's Chalkboard's user avatar
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Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite ...
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Does there exist any term for finite groups with no non-trivial homomorphisms into $\mathbb{Z}/p\mathbb{Z}$ for a fixed prime $p$, or any term related to this property (so that I could write "...
Mikhail Bondarko's user avatar
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Let $G$ be a group. An element $g \in G$ is said to be a test element if any endomorphism $\phi$ of $G$ such that $\phi(g) = g$ is an automorphism. The free group $F_2$ of rank $2$ is generated by $...
Shri's user avatar
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Let $p$ a prime. A finite group is a $p'$-group if its order is prime to $p$. Let $A$ be a finite $p'$-group of automorphisms of a finite $p$-group $G$. Suppose that $A$ is a non-cyclic abelian group. ...
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177 views

If we have profinite groups $G_1,...,G_n$ we can define its free profinite product in the natural way. But this natural definition (similar to the abstract case but in the category of profinite groups)...
Lucas's user avatar
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Let $\mathbb{F}_{p}$ be a finite field of order $p$, $\mathbb{Z}_p$ be the ring of $p$-adic integers and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. For $p\geq 5$, ...
stupid boy's user avatar
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2 votes
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96 views

Let $\mathcal{O}_{L}$ be the ring of integers of a finite extension $L$ of $p$-adic number fields $\mathbb{Q}_{p}$ where $p$ is an odd prime. Let $\mathfrak{m}_{L}$ be the maximal ideal of $\mathcal{O}...
stupid boy's user avatar
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Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$ be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$. We give it the product ...
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Let $L/\mathbb{Q}$ (resp. $K/\mathbb{Q}$) be a Galois extension of rational number field $\mathbb{Q}$ with Galois group $P$ (resp. $H$) where $P$ is a second countable pro-$p$ group and $H$ is a ...
stupid boy's user avatar
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4 votes
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$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers, and let $\frak{m}$ be its maximal ideal Let $\GL_n(\mathcal{O})$ be the group ...
asv's user avatar
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I am interested in the Sylow subgroups of the profinite completion of a free product of finite groups. Is the following naive expectation true ? I assume things like this should be well-known, and am ...
user520733's user avatar
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0 answers
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I do not understand on page 6 of Galois Cohomology from Serre, the comment after exercise 2) part d). He claims that taking G to be the dual of a countably dimensional vector space over $\mathbb{F}_p$ ...
Rodolphe's user avatar
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Let $\mathbb{F}_{p}$ be a finite field of order $p$, and $\mathbb{F}_{p}[[T]]$ be the ring of formal power series over $\mathbb{F}_{p}$. My question is the following: Let $G$ be a closed pro-$p$ ...
stupid boy's user avatar
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2 votes
0 answers
143 views

An infinite profinite group $G$ is called just-infinite if all non-trivial closed normal subgroups of $G$ have finite index. A profinite group is called hereditarily just-infinite if every open ...
stupid boy's user avatar
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0 answers
153 views

Let $p$ be a prime and $G$ be a finitely generated pro-$p$ group admitting a continuous automorphism $\phi$ of finite order relatively prime to $p$. Let $\Phi(G)$ denote the Frattini subgroup of $G$. ...
stupid boy's user avatar
  • 917
1 vote
0 answers
114 views

Let $\mathbb{F}_{q}$ be a finite field of order $q$, and $\mathbb{F}_{q}[[T]]$ be the ring of formal power series over $\mathbb{F}_{q}$. We say that a profinite group $G$ is Noetherian if any closed ...
stupid boy's user avatar
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198 views

Let $p$ be a prime and $\mathbb{Z}_p$ the ring of $p$-adic integers. Let ${\rm PSL}_{2}(\mathbb{Z}_{p})={\rm SL}_{2}(\mathbb{Z}_{p})/\{\pm 1\}$ be the projective special linear group over $\mathbb{Z}...
stupid boy's user avatar
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2 votes
1 answer
156 views

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 $G$...
Lucas's user avatar
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3 votes
0 answers
217 views

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 ...
Alex Byard's user avatar
  • 263
9 votes
1 answer
252 views

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) ...
Alex Byard's user avatar
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6 votes
2 answers
233 views

For a finite $p$-group $G$, let $\mho_i(G)$ denote the subgroup generated by $p^i$-powers of elements of $G$. It is well-known that $\mho_i(\mho_j(G))$ can differ from $\mho_j(\mho_i(G))$ and from $\...
grok's user avatar
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4 votes
1 answer
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I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a ...
Alex Byard's user avatar
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4 votes
0 answers
209 views

For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...
Nobody's user avatar
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1 vote
0 answers
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Let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group of the maximal extension of $ K $ (inside a fixed algebraic closure of $K$) unramified outside $ S $. In ...
Nobody's user avatar
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0 votes
0 answers
84 views

Let $G$ be a topologically finitely generated infinite pro-$p$ group. Suppose that $G$ is not just-infinite. Does the group $G$ always have a maximal topologically characteristic subgroup of infinite ...
stupid boy's user avatar
  • 917
7 votes
1 answer
444 views

$\DeclareMathOperator\SL{SL}$ Let $p>3$ and $G$ be an open subgroup of the special linear group $\SL_2(\mathbf{Z}_p)$ over the ring $\mathbf{Z}_p$ of $p$-adic integers. Suppose that $G$ is ...
trivialquestions's user avatar
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0 votes
1 answer
180 views

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\GL{GL}$Let $G_2(\mathbf{Z}_p):=\begin{pmatrix} 1+p\mathbf{Z}_{p} & \mathbf{Z}_{p}\\ p\mathbf{Z}_{p} & 1+p \mathbf{Z}_{p} \end{pmatrix}$. ...
trivialquestions's user avatar
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0 answers
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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 ...
trivialquestions's user avatar
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