bio | website | |
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location | Université Kasdi Merbah Ouargla, Ouargla, Algeria | |
age | 29 | |
visits | member for | 2 years, 5 months |
seen | Jul 25 at 13:50 | |
stats | profile views | 519 |
"Aujourd’hui, j’exprimerais la leçon que j’ai apprise alors ainsi : "les scientiﬁques", des plus illustres aux plus obscurs, sont des gens exactement comme tous les autres ! Je m’étais complu à m’imaginer que "nous" étions quelque chose de mieux, que nous avions quelque chose en sus - il m’a fallu bien un an ou deux pour me débarrasser de cette illusion-là, décidément tenace !"
Alexandar Grothendieck.
Jul 25 |
comment |
A question on direct limits of finite $p$-groups
Artuto, I'm really thankful (sorry for the delay). |
Jul 25 |
accepted | A question on direct limits of finite $p$-groups |
Jul 25 |
revised |
Torsion in profinite groups
deleted 43 characters in body |
Jul 25 |
revised |
Torsion in profinite groups
edited body |
Jul 25 |
revised |
Torsion in profinite groups
added 707 characters in body |
Jul 22 |
comment |
Torsion in profinite groups
You have mentioned that Zelmanov's solution of RBP, implies that every finitely generated torsion profinite group is finite. I'm not sure about this; it is more safe to say that it implies that every finitely generated profinite group of finite exponent is finite. The result that you mentioned follows from a more general result of Zelmanov (which I'm not sure that is equivalent to the positive solution of the RBP). |
Jul 22 |
comment |
Torsion in profinite groups
-containing $\overline{G^n}$. Pick such a $N$ such that $G/N$ has the maximal possible order. If $M$ is another normal open subgroup containing $\overline{G^n}$, then so is $M\cap N$, if $N$ does not lie in $M$, then $G/(M\cap N)$ has order greater than $|G/N|$, a contradiction. Thus $N$ is contained in every normal open subgroup containing $\overline{G^n}$; so $N=\overline{G^n}$. |
Jul 22 |
comment |
Torsion in profinite groups
@Pablo: If there is a bound on the n's, we may assume that $x^n\in K$, for some $n$ and for all $x$. The subgroup $G^n$ generated by all the $x^n$'s lies in $K$, and so is $\overline{G^n}$. We have only to show that $G/\overline{G^n}$ is finite. First, note that $\overline{G^n}$ is the intersection of open normal subgroups containing it. By the solution of RBP, there is a bound $f(d,n)$ on the orders of $d$-generated finite $p$-groups satisfying the identity $x^n$. If we assume that $G$ is $d$-generated, then $|G/N|$ is bounded by $f(d,n)$, for every normal open subgroup $N$ |
Jul 22 |
revised |
Torsion in profinite groups
added 269 characters in body |
Jul 21 |
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Torsion in profinite groups
As usual, I done a stupid mistake. I'm using trivially to mean "empty". The answer is false. I have to delete it. |
Jul 21 |
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Torsion in profinite groups
Ok, in a compact space, from every family of closed subsets which intersect trivially, one can extract a finite subfamily which intersects trivially. (Please check if I done a mistake in the answer; usually, this is the case) |
Jul 21 |
awarded | Revival |
Jul 21 |
revised |
Torsion in profinite groups
added 86 characters in body |
Jul 21 |
answered | Torsion in profinite groups |
Jul 21 |
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Is there a Noetherian profinite group of infinite rank?
The question is well-known for pro-$p$ groups. You can find it under the form : is every noetherian pro-$p$ group analytique $p$-adic? (See for instance "New Horizons in pro-$p$ Groups", by du Sautoy et Al; Appendix, Problem 1. |
Jul 21 |
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Torsion in profinite groups
No, if you assume that there is a bound on the $n$'s, or if $K$ is normal. I think the first is equivalent to the positive solution of the Restriceted Burnside Problem; the second follows from Zelmanov's Lie algebra analogue of Shirshov's theorem for associative algebras. |
Sep 24 |
awarded | Autobiographer |
Jul 2 |
awarded | Curious |
May 11 |
comment |
A question on $p$-central $p$-groups
@Geoff Robinson: Yes you are right. For $p=2$, one say that $G$ is $2$-central if every element of order dividing 4 is central. $A$ can be taken to be of exponent 4, so Alperin's result can be applied. |
May 10 |
asked | A question on direct limits of finite $p$-groups |