Timeline for Applications for p-Sylow subgroups theorem
Current License: CC BY-SA 3.0
30 events
| when toggle format | what | by | license | comment | |
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| Mar 23, 2023 at 4:25 | answer | added | Chris Gerig | timeline score: 1 | |
| Apr 13, 2012 at 17:37 | history | edited | Beni Bogosel | CC BY-SA 3.0 |
added 2 characters in body
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| Mar 28, 2012 at 5:20 | answer | added | DavidLHarden | timeline score: 4 | |
| Jun 23, 2011 at 3:11 | answer | added | Benjamin Steinberg | timeline score: 21 | |
| May 1, 2011 at 13:42 | answer | added | Greg Kuperberg | timeline score: 14 | |
| May 1, 2011 at 8:03 | answer | added | Geoff Robinson | timeline score: 7 | |
| Apr 5, 2011 at 19:57 | answer | added | maproom | timeline score: 19 | |
| Apr 5, 2011 at 11:15 | comment | added | darij grinberg | @Mariano: This is representation theory of Lie groups. I should have said that I mean group theory rather than representation theory, and that I also mean finite rather than Lie. | |
| Apr 5, 2011 at 11:14 | comment | added | darij grinberg | @Yemon: the question says "spectacular". I normally wouldn't consider an application to group theory itself spectacular, as the Sylow theorem pretty much suggests itself when you do group theory (after all, it's a standard trick that if one doesn't understand some object, one tries to "reduce it to primes"). | |
| Apr 5, 2011 at 11:12 | comment | added | darij grinberg | ... to begin with. I know of one exception: the fundamental theorem of algebra. I would like to know more. And as for the theorems more advanced than Sylow's (Frobenius complement, Frattini argument, Feit-Thompson), I don't know of any exception at all. | |
| Apr 5, 2011 at 11:11 | comment | added | darij grinberg | OK, I didn't expect people to swarm on my formulation "group theory". What I meant is FINITE group theory. I think that any advanced theorems in FINITE group theory, where advanced begins with Sylow (Cauchy is ok still, the proof is short) can be avoided pretty much everywhere else. As always, such a claim is an invitation for proving the opposite. I wouldn't be surprised if some results about finite orbits in dynamics have a use for finite groups, and some representation theory (obviously), but I wouldn't expect Sylow to be helpful in proving a theorem which is NOT about finite groups ... | |
| Apr 5, 2011 at 9:59 | vote | accept | Beni Bogosel | ||
| S Apr 5, 2011 at 9:59 | vote | accept | Beni Bogosel | ||
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| Apr 5, 2011 at 9:59 | vote | accept | Beni Bogosel | ||
| S Apr 5, 2011 at 9:59 | |||||
| Apr 5, 2011 at 9:58 | vote | accept | Beni Bogosel | ||
| Apr 5, 2011 at 9:59 | |||||
| Apr 5, 2011 at 2:42 | comment | added | Yemon Choi | Also, in response to one of Darij's early comments: the question quite clearly asks for "applications of Sylow's theorem in group theory and other fields of mathematics (which are of course related to groups)". I think it is stretching things to interpret it as your question "why should I care for Sylow's theorems if I am not a group theorist". | |
| Apr 5, 2011 at 2:39 | comment | added | Yemon Choi | Further to Pete's remark: it is possible to avoid things, but it may not be healthy. | |
| Apr 5, 2011 at 2:34 | comment | added | Yemon Choi | I think the question could be improved by clarifying what counts as "spectacular" - this kind of judgment is something on which many mathematicians change their minds over the years. | |
| Apr 5, 2011 at 2:24 | comment | added | Mariano Suárez-Álvarez | Well... I took a short course on harmonic polynomials and other special polynomials---beautiful analysis---and at some point, talking to the speaker, I asked him something related to the orthogonal group. I was pretty amazed at finding out that she was not at all aware of the fact that during the previous 4 hours she had been decomposing the representation of $SO(n)$ on $L^2(S^{n-1})$, that her polynomials where special because they diagonalized certain subgroups and so on! On can very well do things very well which could be expressed using groups and think about them in other ways. | |
| Apr 5, 2011 at 2:24 | answer | added | Chuck | timeline score: 23 | |
| Apr 5, 2011 at 1:47 | comment | added | KConrad | I know a well-respected probabilist who has said he has never had any reason to use the material from his abstract algebra courses, so that is one example of a successful mathematician who does not need to know about finite groups. I do not think it is correct to say one can avoid group theory in most other fields of mathematics, but that's not the same as saying one can avoid most of group theory in other fields of mathematics. If you pull a massive book on group theory off the shelf (as a model for "most of group theory"), is it all really needed in other areas? Hmm... | |
| Apr 5, 2011 at 1:16 | comment | added | Pete L. Clark | "While group theory can be beautiful, one can pretty much avoid most of it in all other fields of mathematics..." I confess that it seems likely to me that no 22 year-old could have a faithful impression of "all other fields of mathematics". Rather, remarks like this strike me as advertisements of one's parochialism. The ubiquity of group theory in mathematics is guaranteed (e.g.) by the ubiquity of category theory and the fact that the automorphisms of an object in a category form a group. There are other reasons! It is a strange mathematician who has zero interest in finite groups. | |
| Apr 5, 2011 at 1:13 | answer | added | GH from MO | timeline score: 10 | |
| Apr 5, 2011 at 0:57 | answer | added | KConrad | timeline score: 30 | |
| Apr 5, 2011 at 0:07 | comment | added | darij grinberg | As I said "... if I am not a group theorist". While group theory can be beautiful, one can pretty much avoid most of it in all other fields of mathematics, so the question still stands. | |
| Apr 4, 2011 at 23:39 | comment | added | user6976 | I do not know any non-trivial result about the structure of finite groups that does not require Sylow theorem. So if you think finite groups are important, then so is the Sylow theorem. | |
| Apr 4, 2011 at 22:36 | comment | added | darij grinberg | Also, Johannes, I disagree. Most books do not give "spectacular" applications of Sylow's theorem, but just some applications to theorems which don't look any more interesting than Sylow. The question, in the form "why should I care for Sylow's theorems if I am not a group theorist", is something I thought about asking some time ago. I am definitely NOT voting to close this. | |
| Apr 4, 2011 at 22:35 | comment | added | darij grinberg | I think there is a well-known application to proving the fundamental theorem of algebra with almost no analysis needed. | |
| Apr 4, 2011 at 19:39 | comment | added | Johannes Hahn | Just look in any book on basic group theory. This is not the right kind of question for mathoverflow. Such a question would be more appropriate in math.stackexchange.com for example. | |
| Apr 4, 2011 at 19:36 | history | asked | Beni Bogosel | CC BY-SA 2.5 |