Timeline for non-continuous inverse Galois problem
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 5, 2012 at 20:40 | vote | accept | Hugo Chapdelaine | ||
| Jun 5, 2012 at 7:03 | history | edited | Will Sawin | CC BY-SA 3.0 |
added 76 characters in body
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| Jun 5, 2012 at 5:33 | comment | added | Benjamin Steinberg | I'm pretty sure it falls out the Segal and Nikolov results that the abstract commutator subgroup is closed for finitely generated profinite groups. I believe it is false in general and that counterexamples can be found in the book of Ribes and Zalesskii | |
| Jun 5, 2012 at 5:18 | comment | added | Kevin Ventullo | @Will: This only shows the set of commutators is closed. Why should the commutator subgroup be closed? | |
| Jun 5, 2012 at 1:15 | comment | added | Benjamin Steinberg | Andreas's answer to mathoverflow.net/questions/80966 shows that a compact group cannot have a non-trivial discontinuous homomorphism to any finitely generated torsion-free group, in particular not to Z. | |
| Jun 5, 2012 at 0:54 | comment | added | Will Sawin | @Kevin: The limit of commutators is the commutator of the limit, and the space is sequential because it is first-countable. @Ben: Cool argument. | |
| Jun 5, 2012 at 0:02 | comment | added | Ben Wieland | No profinite group can discontinuously surject to $\mathbb Z$. If so, choose a splitting and take the closure of the image. This is a commutative profinite group surjecting to $\mathbb Z$. In fact, it is a quotient of the free profinite group on one generator, $\hat {\mathbb Z}$, and Will shows this group doesn't map to $\mathbb Z$.....It is worth mentioning that on a finitely generated profinite group, the topology is intrinsic. | |
| Jun 4, 2012 at 23:21 | comment | added | Yiftach Barnea | @Kevin: from number theortical and Galois theory points of view the abstract group might not be interesting. However, there is an increasing intrest amongst group theoriests in the abstract structure of profinite groups. So yes it could be of some interest. You could ask Nik Nikolov a bit more about it. | |
| Jun 4, 2012 at 22:28 | comment | added | Kevin Buzzard | Will: for Q1 are you saying/implying that the abstract commutator subgroup is closed? Why is this? Also surely the answer to Q2 is "yes", at least if you believe the inverse Galois problem. That said, this question seems very pathological to me. I can't really see any logic in considering a Galois group without its topology -- somehow the topology is what makes the Galois group worth studying. Do people really care about just the abstract group structure? | |
| Jun 4, 2012 at 22:16 | history | answered | Will Sawin | CC BY-SA 3.0 |