Talk:Classification of finite simple groups
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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)
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)
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)
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)
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- 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.
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- 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)
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
