# Tag Info

Accepted

• 52.1k
Accepted

### Sylow subgroups of abelian profinite groups

This is Proposition 2.3.8 of Ribes and Zaleskii - Profinite groups (second edition). (I originally gave references specifically for the finer structure of profinite Abelian groups, but assuming ...
• 8,059

### Is there a residually finite non-elementary hyperbolic group whose profinite completion is boundedly generated?

I'm fairly certain that no example is known. Of course, it's a famous open problem whether every hyperbolic group is residually finite. This turns out to be equivalent to many other questions about ...
• 22.8k
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### Avoiding countable subgroups of a group homeomorphic to the Cantor space

Yes, there exists. Since $G$ is profinite, we may write it as $G=\lim_{n\in\mathbb{N}} G_n$, where $G_n$ are finite and the projections $\pi_n:G\to G_n$ are onto. Since $H$ is countable, we may write ...
• 3,011
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### Applications of Lubotzky's linearity theorem?

According to Alex himself, this theorem is practically useless. It does not mean that it can't be applied, for instance when you have a group with assumptions that it has many quotients in some ...

### Subgroups of hyperbolic groups

If there's a non-residually finite hyperbolic group, then the answer is no. By a result of Kapovich-Wise, in this case there is a hyperbolic group $H$ whose profinite completion is trivial. Then ...
• 62.1k
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• 1,569
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### Do free profinite groups satisfy Howson's theorem?

The following example shows that the answer is no. Let $p$ and $q$ be two different primes. First I want to construct two generated profinite group $G=A\ltimes H$ isomorphic to semi direct product of ...
• 1,298
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• 32.8k
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### Dessins d'enfants and absolute Galois group

For the sake of getting this off the unanswered stack... Please refer to the following reference: Guillot, P. An elementary approach to dessins d'enfants and the Grothendieck-Teichmüller group, ...
• 10.9k
This has already been answered in the comments, but perhaps you can see it more clearly like this. Take the isomorphism $\mathrm{Gal}(\bar{\mathbb{Q}}|\mathbb{Q}) \cong \varprojlim(K|\mathbb{Q})$ as ...
To the question in the title: Not in general, e.g the action of (an infinite profinite group) $G$ on itself by left multiplication is continuous with infinite orbits. However, note that the action of ...