Talk:Classification of finite simple groups
From Wikipedia, the free encyclopedia
 |
|||||||||||||||||||||||
| Mathematics rating: | B Class | Top Priority | Field: Algebra | ||||||||||||||||||||
|
|||||||||||||||||||||||
Contents |
[edit] Classical simple groups
Would it be correct to characterize the "classic" simple groups as those which can be represented as products of matrices over C? Chas zzz brown 23:31 Feb 15, 2003 (UTC)
- No, the classic simple groups are certain quotients of linear groups over finite fields. AxelBoldt 20:07 14 Jun 2003 (UTC)
- Chas, you're thinking of the simple Lie groups. -- Toby Bartels 04:05, 24 Aug 2003 (UTC)
[edit] Sporadic groups
Anyone want to write a bit about each of the sporadic groups? :-) -- Timwi 15:34 4 Jul 2003 (UTC)
Also, the article seems to imply that these 26 sporadic simple groups are the only ones that don't fit into the other four categories. Has this been proven? -- Timwi 15:34 4 Jul 2003 (UTC)
Yes; indeed, that's precisely the difficult content of the enormous proof. To show that each of these things is a simple group in the first place is much easier; to show that you've left nothing out is the hard part. -- Toby Bartels 04:05, 24 Aug 2003 (UTC)
Actually the proof of the uniqueness of the 26 sporadic groups is still a debated issue. For instance the uniqueness proof of the Thompson group is flawed, although a new proof of the uniqueness will be published in the coming months. Since the paper hasn't been published yet I felt it wouldn't be appropriate to add to the article, but just FYI there are still some significant holes in the proof. TooMuchMath 18:44, 15 April 2006 (UTC)
[edit] Unitary and Lie links
In the list of groups: links to "unitary" and "Lie" do not seem to be appropriate. I expected that these links point to where unitary groups over _finite_ fields and groups of Lie type are discussed. Instead of these I have found unitary groups over R and C and real and complex Lie groups.
Answer: The unitary link should be to special unitary group; and, yes, better to have there the case of finite fields. You can see from the page history that the restriction to the complex field case is quite recent. For some reason the development over any field F (with automorphism *) never happened. As you say, this link is therefore not too helpful at present.
In the case of groups of Lie type, we are still waiting for a Chevalley group article; so I don't really see that the topic can be done justice, right now.
Charles Matthews 09:26, 14 May 2004 (UTC)
[edit] Mathieu group dates
Is anyone able to verify the statement about the date of discovery of the Mathieu groups? I have been told that M11 and M12 were discovered quite a few years before M22, M23 and M24. A google search suggests a date of 1873, but it'd be nice to have some confirmation from someone.
--Huppybanny 21:35, 16 May 2004 (UTC)
- Answer: Mathieu published papers in 1861 and 1873. I have not seen them, but I understand that M11 and M12 appeared in the 1861 paper and the 3 large groups in the 1873 paper. They are cited in Dixon & Mortimer's "Permutation groups". Scott Tillinghast, Houston TX 05:19, 22 July 2006 (UTC)
-
- E. Mathieu, 'Mémoire sur l'étude des functions des plusieurs quantités, sur le manière de les former et sur les sustitutions qui les laissent invariables', J. Math. Pures Appl. (Liouville) (2) 6 (1861), 241-323.
-
- E. Mathieu, 'Sur la function cinq fois transitive de 24 quantités', J. Math. Pures Appl. (Liouville) (2) 18 (1873), 25-46.
Scott Tillinghast, Houston TX 19:46, 25 July 2006 (UTC)
[edit] Infinite simple groups?
Is there any such thing as an infinite simple group, and if not, then would it make sense to merge this page into Simple group? -℘yrop (talk) 18:51, Dec 12, 2004 (UTC)
Apparently there is: see Thompson groups. -℘yrop (talk) 06:34, Jan 14, 2005 (UTC)
There are many infinite simple groups. Having this page on the finite simple case is more than justified. Charles Matthews 21:13, 10 Mar 2005 (UTC)
[edit] HJ story
The explanation that HJ stands for Hall-Janko should be left to the group's own page. Each group has its own story.--192.35.35.36 20:18, 10 Mar 2005 (UTC)
[edit] Unifying theme
Is there really not a "convincing unification of the sporadic simple groups?" All but 6 are found in the Monster. The Mathieu groups are a niche within Conway 1, which is a niche within the Monster. The expansion of the Mathieu groups (binary Golay code) to the Conway groups (Leech lattice) seems natural enough - both relate to peculiarities of 24-dimensional spaces. I suppose the Monster and the Griess algebra represent a similar expansion. [Scott Tillinghast, Houston TX] 12:57 15 Mar 2006
[edit] could someone give a "basic idea" of the original classification program?
It would be nice if someone that knows something about the original classification could write a paragraph about the technical side of the program. I heard Borcherds say something to the effect of "you look at centralizers of involutions" and then explain some kind of recursion principle, but I didn't understand it well enough to write a meaningful/accurate summary. Kinser 17:54, 22 February 2007 (UTC)
Partial answer: It would need more than a paragraph, and would be hard to keep non-technical, as you will see below, but something could be attempted. The following is an over-simplification, far from comprehensive, and is too imprecise to put in the article, but gives some flavor: The 1956 result of Brauer-Fowler showing that there are only finitely many simple groups with a centralizer of an involution ( element of order 2) of given order led to numerous results to characterize simple groups by specifying a known structure for the centralizer of an involution, several of which led to new sporadic simple groups, notably in work of Z.Janko. This, combined with the 1963 Odd order theorem of Feit and Thompson, and the Brauer-Suzuki theorem ( approx. 1958) showing that no finite simple group has a (generalized) quaternion Sylow 2-subgroup, concentrated attention on elementary Abelian 2-subgroups of simple groups. J.G. Thompson's classification of N-groups (which covered the classification of minimal finite simple groups) introduced new techniques which were later refined to what became known as the "signalizer functor method" in one direction, and "failure of factorization" techniques in another. Signalizer methods were particularly effective with elementary Abelian 2-groups of order at least 16, slightly less so with elementary Abelian subgroups of order 8 (and not applicable with Klein four groups). Fortunately, character-theoretic methods (sometimes using modular representation theory) as developed by Brauer and others, were well suited to dealing with groups having no elementary Abelian subgroup of order 8, eg groups with dihedral Sylow 2-subgroups.In the presence of large enough elementary Abelian 2-subgroups, techniques such as signalizer functor methods essentially subdivided the later stages into two cases: one case (odd-type) where the centralizer of some involution resembles such a centralizer in a group of Lie type over a field of odd characteristic, and the "characteristic 2-type" case, where all involutions have centralizers resembling those in groups of Lie type in characteristic 2. Roughly speaking, the goal was to identify the former as groups of Lie type in odd characteristic, and the latter as groups of Lie type in characteristic 2, with known exceptions (eg many sporadic groups have characteristic 2-type). In all these cases, work of Michael Aschbacher led to much progress. The characteristic 2-type case proved to be by far the most difficult, and required splitting into various cases, and using signalizer methods for primes other than 2. The quasithin case was one of the subdivisions and here generic methods were not available, and the power of odd signalizer functor methods was limited. Other names ( not already mentioned) playing significant roles include: Bender,Fischer,Foote,Glauberman,Goldschmidt,Gorenstein,Greiss,Harada,Lyons,Mason,McBride,Solomon,: Stellmacher,Stroth,Timmesfeld. The "amalgam method", introduced by Goldschmidt in the later stages of the program plays a larger role in the second generation program, and is intended to play an even greater role in the third generation program (for all primes, not just the prime 2, in the 3G case). BTW: The names of Lyons and Solomon should figure more prominently on the current page, especially discussing revisionism and second generation ( and I am neither of them). BTW again: An elementary Abelian 2-group is a finite group all of whose non-identity elements have order 2. Messagetolove 21:04, 7 May 2007 (UTC)
[edit] History of finite simple groups
As a humble lay reader, I can recommend Symmetry and the Monster by Mark Ronan (Oxford University Press, 2006, ISBN 978-0-19-280723-6) for anyone seeking a historical overview of the discovery of the 26 exceptional finite simple groups and the initiation of the classification programme.--Calabraxthis 08:31, 1 December 2007 (UTC)
