Let $K$ be a finite extension of the $p$-adic numbers. $G_K$ be its absolute Galois group and $I_K$ the inertia subgroup. Are finite index subgroups of $I_K$ closed in its profinit …

Let $P_n$ denote the pro-$p$ completion of $F_n$ the free group of rank $n$. Given a (abstract) group homomorphism $$ \phi:P_n\rightarrow G $$ where $G$ is a discrete group. Is $\ …

Does every profinite group arise as the étale fundamental group of a connected scheme? Equivalently, does every Galois category arise as the category of finite étale covers of a c …

Assume $X$, $E$ and $G$ are topological groups and $1\to X\to E\to G\to 1$ a short exact sequence of continuous group homomorphisms. Under which of these conditions is $E$ a profin …

I've been reading about profinite groups and have encountered the notion of strong completeness. I.e. that a profinite group $G$ is strongly complete if it is isomorphic to it's pr …

If $G_1,\dots,G_n$ are discrete groups, and $G=G_1 \ast G_2 \ast \dots \ast G_n$ is their free product, then for $g \in G_i$ sean as an element of $G$, it is clear the centralizer …

Assume $A, B, C$ are profinite groups and $0\to A\to B\to C\to 0$ is an exact sequence of continuous maps. Which of the following assertions follows?: (i) the subspace-topology in …

Let $G$ a locally profinite not compact group and we suppose that $G$ have maximal compact subgroups, for example $G$ reductive group over p-adic field. We call projective Tower o …

A pro-$p$ group of finite subgroup rank has an open subgroup $P$ that is uniformly powerful, meaning that $[P,P]$ is contained in the group generated by $2p$-th powers in $P$, and …

I am looking for an example of the following situation: $G$ is an infinite profinite group, with a dense normal subgroup $N$. However $N$ does not contain any non-trivial closed …

Does anyone know of any interesting examples of an infinite abelian $p$-group which is uncountable? By non-interesting here I mean the direct sums of cylic and quasi-cylic groups, …

Let $G$ be a profinite group which fits in the following short exact sequence: $$ 1\rightarrow N\rightarrow G \rightarrow K\rightarrow 1 $$ Assume that $N$ is a pro-$p$ group and t …

Is there a profinite group $G$ which is not its own profinite completion? Surely not, I thought. But upon looking into it, I found that there is a special name given to a $G$ …

Following the discussion I have with Yves Cornulier in the following question http://mathoverflow.net/questions/98696/finiteness-theorems-for-profinite-groups, I would like to ask …

The following questions popped out while I was preparing a course on profinite groups. Closed subgroups of free profinite groups are not necessarily profinite free (e.g. the p-sy …