Timeline for Character-theoretic proofs of Burnside's theorem & Frobenius's theorem: conservation of difficulty?
Current License: CC BY-SA 4.0
31 events
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| Nov 10 at 11:51 | comment | added | Peter Mueller | ... linear substitutions. There is accordingly in the present edition a large amount of new matter. | |
| Nov 9 at 13:19 | comment | added | Peter Mueller | ... and from the second edition 1911: Very considerable advances in the theory of groups of finite order have been made since the the appearance of the first edition of this book. In particular the theory of groups of linear substitutions has been the subject of numerous and important investigations by several writers; and the reason given in the original preface for omitting any account of it no longer holds good. In fact it is now more true to say that for further advances in the abstract theory one must look largely to the representation of a group as a group of ... | |
| Nov 9 at 13:17 | comment | added | Peter Mueller | From the first edition 1897 of Burnside's book: It may then be asked why [...] a considerable space is devoted to substitution groups; while other particular modes of representation, such as groups of linear transformations, are not even referred to. My answer to this question is that while, in the present state of our knowledge, many results in the pure theory are arrived at most readily by dealing with properties of substitution groups, it would be difficult to find a result that could be most directly obtained by the consideration of groups of linear transformations. | |
| Nov 9 at 12:32 | comment | added | Geoff Robinson | @SteveD : It;s not just speculation on the part of Marty Isaacs' part: Burnside says in the preface to the second edition of his book on Group Theory (words to the effect that) that he did not include character theory in the first edition, but that he now realised that character theory was more pertinent to group theory., | |
| S Oct 10 at 17:04 | history | bounty ended | CommunityBot | ||
| S Oct 10 at 17:04 | history | notice removed | CommunityBot | ||
| Oct 3 at 20:07 | comment | added | Steve D | In Finite Group Theory by Isaacs, he gives the "group-theoretic" proof of Burnside's theorem, as well as some (historical) context. He posits that finding his character-theoretic proof may have completely reversed Burnside's thinking on the usefulness of character theory! | |
| Oct 3 at 18:42 | answer | added | Garrett Figueroa | timeline score: 6 | |
| Oct 3 at 0:25 | comment | added | Steve D | @LSpice: it's the size of the conjugacy class. | |
| Oct 3 at 0:25 | comment | added | LSpice | @PeterMueller, what is the length of a conjugacy class? | |
| Oct 2 at 19:28 | comment | added | Joshua Grochow | Timothy Chow's comment applies also to Mark Schultz-Wu's comment. The non-probabilistic proofs are harder, but often yield more information than mere existence. There is a reason so many problems in extremal combinatorics and CS are of the form "We know this exists by a probabilistic argument, but we need to find one concretely/constructively." | |
| Oct 2 at 19:12 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| S Oct 2 at 15:14 | history | bounty started | Nicolas Banks | ||
| S Oct 2 at 15:14 | history | notice added | Nicolas Banks | Draw attention | |
| Sep 27 at 12:48 | comment | added | Timothy Chow | I don't understand the group-theoretic proof of Burnside's theorem, so I don't know if the following general observation applies, but if Proof 1 seems a lot harder than Proof 2, then it is often because Proof 1 actually yields a stronger result. For example, to prove that some quantity is a positive integer, it is enough to prove that it is the dimension of some vector space. But giving a combinatorial interpretation of the integer typically yields an explicit basis for said vector space. So, perhaps the group-theoretic proof of Burnside's theorem yields "extra information"? | |
| Sep 27 at 6:40 | comment | added | Fedor Petrov | @WillSawin well, for me even the idea of the notion "an event is independent on the subalgebra of some other events" is cumbersome and seemingly artificial without probabilistic language. But I agree that it does not use measure theory foundations. There are, still, continuous methods in combinatorics which do use it essentially, like ergodic theory methods (Furstenberg and beyond), or graph limits. In these areas there is a notion of "elementary proof", like in analytic number theory, where the role of complex analysis is played by measure theory and functional analysis | |
| Sep 26 at 13:09 | comment | added | Peter Mueller | Note that Burnside's Theorem is an easy consequence of the following: A finite non-abelian simple group does not have a conjugacy class of primepower length $> 1$. In fact the latter is what the character theoretic proof of Burnside's theorem shows. Nevertheless, as far as I know, this latter fact does not have a character-free proof yet. | |
| Sep 26 at 10:53 | comment | added | Will Sawin | But I went carefully through the proof of the Lovasz local lemma and replaced the probabilities and conditional probabilities with ratios of the sizes of sets and it seems not much more cumbersome. Joining two vertices with an edge with probability $n^{-\theta}$ can often be replaced by considering graphs with exactly $n^{2-\theta}$ edges, which admittedly can be significantly more cumbersome. But this has nothing to do with foundations. | |
| Sep 26 at 10:48 | comment | added | Will Sawin | @FedorPetrov I'm not talking about the probabilistic view but rather the probabilistic foundation. The point I was trying, perhaps badly, to make is that the idea of thinking probabilistically, of considering what proportion of objects (maybe with some weighting) satisfy a property when asked whether an object satisfying the property exists, is what's crucial, and the foundations of probability theory (e.g. measure theory) are usually irrelevant. The point is that the difficulty is not hiding in the foundations (or in the proof of the Lovasz local lemma) but in finding the right idea. | |
| Sep 26 at 1:59 | comment | added | Fedor Petrov | @WillSawin even such simple and common thing as independent non-uniform Bernoulli ("join every two vertices by an edge with probability $n^{-\theta}$") is not about sizes but about weighted sums, which look involved without probabilistic view and natural with it. Reducing more advanced tools like entropy, or Lovasz local lemma, to counting the sizes of sets seems practically impossible for me. | |
| Sep 26 at 0:42 | comment | added | Will Sawin | The probabilistic method does not really use the probabilistic foundation. Most (almost all?) applications in combinatorics can be expressed as arguments about counting the sizes of certain sets. | |
| Sep 25 at 22:46 | comment | added | Mark Schultz-Wu | @NicolasBanks Constructing good codes has been a topic of study for getting close to 100 years (that has even had industry backing). Near-optimal codes (even ones that admit a compact description, via having a "linear" structure) can be easily constructed using randomized constructions. These have some practical difficulties (decoding them is slow), but showed impossibility results are (close to iirc) tight. Constructing such codes without the probabilistic method is much harder. | |
| Sep 25 at 22:36 | comment | added | Nicolas Banks | @AndyPutman the reference is much appreciated. I'm not trying to make more of Burnside's Theorem than is warranted, but I am still surprised at the utility of character theory that it showcases. | |
| Sep 25 at 22:35 | comment | added | Nicolas Banks | @MarkSchultz-Wu do you have any particular examples? I'm aware of uses for probability theory e.g. in graph theory but not familiar enough to gauge the extent that significant results are hiding in the probabilistic foundation. | |
| Sep 25 at 22:34 | comment | added | Dave Benson | In the proof of the classification of finite simple groups, it's just as remarkable how character theoretic methods (including modular character theoretic methods such as Glauberman's proof of the $Z^*$ theorem) give out long before the main structural methods take hold. In the end, I think the best approach is to use whatever methods you can lay your hands on for a first proof, and then try to see what's really necessary or best later on. | |
| Sep 25 at 22:22 | comment | added | Mark Schultz-Wu | In general, I don't know if "conservation of difficulty" is particularly true. There are many easy examples (say the probabilistic method for one) where you can take something that is very hard with other techniques, and provide a very simple proof of it that does not really hide non-trivial math. | |
| Sep 25 at 22:16 | history | edited | LSpice | CC BY-SA 4.0 |
No *known* such proof
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| Sep 25 at 22:16 | comment | added | Andy Putman | I’m not sure that Burnside’s theorem is as big a deal as you make it out to be. As far as why character theory is so effective, you’ve got to construct a normal subgroup out of something, and representations are a natural thing to look at (especially when you don’t have another obvious source of homeomorphisms to look at). I wrote an account of the char theory proof Burnside’s theorem that tries to bring out the actual mechanism in a motivated way here: www3.nd.edu/~andyp/notes/BurnsidePQ.pdf | |
| Sep 25 at 22:16 | history | edited | LSpice | CC BY-SA 4.0 |
No *known* such proof
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| Sep 25 at 22:13 | comment | added | Sam Hopkins | There was a lot of effort put into finding an "elementary" proof of the prime number theorem (one avoiding the use of complex numbers), but then when one was found by Selberg/Erdős, not much came of it. It could just be that the character theoretic proof of Frobenius's theorem is the "right" one. | |
| Sep 25 at 22:03 | history | asked | Nicolas Banks | CC BY-SA 4.0 |