# Questions tagged [profinite-groups]

For questions regarding profinite groups and their properties.

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### Infinite Galois Correspondence on Ramakrishnan & Valenza

I'm reading the book Fourier Analysis on Number Fields by Ramakrishnan & Valenza (Theorem 1-20 on page 34). I'm a little confused by the discussion of infinite Galois theory. Let $K/F$ be Galois (...
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### Discrete G-module iff union over submodules fixed by open subgroups

I am working through Weibel's homological algebra and got stuck on a seemingly simple exercise, 6.11.6. We say that a $G$-module $A$ is discrete if the $G$-action is continuous when $A$ is endowed ...
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### Profinite groups with specific property

I saw that a topological group $G$ is profinite if and only if it is compact, Hausdorff and totally disconnected. In such groups, an open subgroup is closed but not vice-versa. From this, I came to a ...
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### Citations to published literature for the profiniteness of Inn(G) for G a profinite group

A recent question asked whether the inner automorphism group of a profinite group is profinite, and received an answer which was mathematically unimpeachable: yes because (1) the center is always a ...
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### Action of profinite groups on primes

Let $G = \varprojlim G_i$ be a profinite group, $A$ a commutative ring with a $G$-action. Suppose furthermore that $A = \bigcup A^{U_i}$, where $U_i$ range over the open normal subgroups of $G$. Is it ...
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### The semidirect product of profinite groups as a surjective inverse system of finite groups

Let $G$ be a second countable profinite group. Then $G$ can be written as a denumerable projective limit $\varprojlim_{i}(\cdots \to G_i\to G_{i-1}\to \cdots )$, where the $G_i$'s are finite and the ...
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### Is a profinite free product of pro-$p$ groups a pro-$p$ group?

I have a question. Maybe it be trivial, but I'm cannot conclude nothing yet. Suppose that we have a free profinite product $$G = G_1 \amalg \cdots \amalg G_n.$$ By definition in Ribes-Zalesskii $G$ is ...
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