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A nomadic postdoc, currently in Australia.
6h

comment 
quasiprimitive unsoluble groups
This paper is probably a good place to start: researchgate.net/publication/… 
Jul 15 
comment 
Products of subgroups that generate a finite group
There are also ZappaSzep products, that is, groups of the form $AB$ for subgroups $A$ and $B$ such that $A \cap B$ is trivial. See for instance math.stackexchange.com/questions/107781/… 
Jul 14 
asked  Separation of topological group elements by invariant neighbourhooods 
Jul 13 
answered  Infimum of two group topologies 
Jul 13 
comment 
Reading Papers in a Language you don't Speak
Google Translate is a poor source of definitions for technical jargon, but Wikipedia is not bad. You should also make an effort to learn standard expressions, like "Soit... Alors..." and "genau dann, wenn...". Thankfully there are not many expressions to learn, as mathematical writing tends to be very 'stiff' and formulaic compared to ordinary prose. 
Jul 12 
comment 
Products of subgroups that generate a finite group
We have $H_1H_2 = H_1H_2/H_1 \cap H_2$, so you can determine whether $G = H_1H_2$ just by looking at orders of subgroups. Also, knowing how many steps it takes to cover the commutator group $[H_1,H_2]$ will give a good estimate, since $G = [H_1,H_2]H_1H_2$. 
Jul 12 
revised 
Locally compact vs. compactly generated in group theory
Answered original question 
Jul 7 
asked  The set of (property) elements of a locally compact group is closed 
Jul 7 
answered  What are the best settings for the large scale geometry of locally compact groups? 
Jul 2 
awarded  Inquisitive 
Jul 2 
awarded  Curious 
Jun 17 
awarded  SelfLearner 
Jun 15 
comment 
Classic applications of Baire category theorem
The Baire category theorem is very useful in topological group theory, because a subgroup either is open or has empty interior, and because there are still plenty of interesting open problems about topological groups that are locally compact and/or Polish. 
May 28 
comment 
Are norms intrinsically $\mathbb{R}$valued?
I think what makes $\mathbb{R}$ special is that it is the largest Archimedean ordered group, so it naturally turns up in any sort of Archimedean norm. 
May 14 
comment 
Interesting examples of generic behavior of mathematical objects being either unreasonably structured or simply unreasonable
Not sure if this counts as an example: if you take a random $k$element subset of the free profinite group $F$ on $d \ge 2$ generators, it will almost always be contained in a proper closed subgroup of $F$, for any $k$. What's interesting is that this is not the case, for instance, for the free prosoluble group on $d$ generators (provided $k$ is large enough). 
May 13 
answered  Locally compact vs. compactly generated in group theory 
May 13 
comment 
Unconventional types of induction
Gerry Myerson: If your base case was instead to prove P(n) for infinitely many n (in a way that does not actually specify which n, e.g. 'by compactness, there is a subsequence on which f(n) converges...'), that ought to count as an unconventional induction. 
Apr 29 
comment 
Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?
Proving almost all finite groups are 2groups looks hard, but perhaps it is reasonable to assert that almost all finite groups are soluble, since nonabelian simple groups are extremely sparse in the class of finite groups and don't admit many extensions. 
Apr 28 
comment 
Automorphisms of profinite groups
Yves Cornulier's comment answers the question for $n$ coprime to $p$: indeed, $\mathrm{Aut}(F)$ is pro$p$by$\mathrm{GL}(d,p)$, so $F$ has an automorphism of order $n$ if and only if $n$ divides $\mathrm{GL}(d,p)$. The next question to ask is whether $F$ has automorphisms of order $p^k$ for arbitrarily large $k$. 
Apr 27 
revised 
Coprime automorphisms of finitely generated pro$p$ groups
Added sufficient condition 