Talk:Classification of finite simple groups

message: 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)
thread_title: Classical simple groups

message: 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)
thread_title: Sporadic groups

message: 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)
thread_title: Unitary and Lie links

message: 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)
thread_title:  Infinite simple groups? 

message: 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)
thread_title: HJ story

message: 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)
thread_title:  could someone give a "basic idea" of the original classification program? 

message: du Sautoy's book contains much that is incorrect. Both his and Ronan's are popular but the latter has a serious
aspect. I recommend Thompson's Carus monograph: From error-correcting codes to simple groups. [It is not
John Thompson.]
John McKay24.200.155.110 (talk) 07:50, 14 August 2008 (UTC)
thread_title: History of finite simple groups

message: <"As of 2005..."> Can anyone please update the status of the second-generation classification? Felix Hoffmann (talk) 06:52, 6 February 2010 (UTC)
replies: text: The status as of November 2012 is addressed here, with an indication that it still had not changed as of February 2014. Robin S (talk) 22:05, 20 July 2015 (UTC)
thread_title:  Status of the second-generation classficication 

message: The article says:
:There has been some controversy as to whether the resulting proof is complete and correct, given its length and complexity
However, it seems that book doesn't see the proof as controversial. I am not even remotely an expert in group theory, but even if there is indeed some controversy, it'd be good to explain why and reference.
Tony (talk) 21:54, 26 October 2010 (UTC)
replies: text: Well Peter J. Cameron wrote in his 1999 book Permutation Groups on p. 108: "It is quite impossible for a layman to judge whether a complete proof of the theorem currently exists." 86.127.138.67 (talk) 14:07, 6 April 2015 (UTC)

text: Meanwhile, Wilson notes in the preface of his 2009 book The Finite Simple Groups (GTM 251) that "With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classification Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete." Those two citations "[12, 13]" refer to the two-volume book The Classification of Quasithin Groups. 86.127.138.67 (talk) 02:08, 11 April 2015 (UTC)
thread_title:  Controversy? 

message: The article says that "Otto Hölder proves that the order of any simple group must be a product of at least 4 primes, and asks for a classification of finite simple groups." If I'm reading this correctly and not being stupid the claimed theorem is false (for example S_5 or Z/pZ for any prime p). There's presumably some additional condition needed. What is missing? JoshuaZ (talk) 21:01, 9 February 2011 (UTC)
thread_title:  Holder's result seems wrong 

message: The term "2-rank" is used frequently in this article, but not defined.  Does anyone know precisely what it means?  Is it related to rank of a group? Jowa fan (talk) 07:08, 10 May 2011 (UTC)
replies: text: The 2-rank of a group G is the maximum number of generators of an Abelian 2-subgroup of G.  —Preceding unsigned comment added by Messagetolove (talk • contribs) 07:16, 10 May 2011 (UTC)
replies: text: Thanks.  Is p-rank defined analogously?  Is there a standard textbook which contains this definition? Jowa fan (talk) 08:11, 10 May 2011 (UTC)
replies: text: Yes, p-rank of a finite group is the largest rank of an elementary abelian p-subgroup.  The sectional p-rank is the largest rank of an elementary abelian p-section (quotient of a subgroup).  Sometimes the sectional rank is better for induction as it can only go down, but the plain rank can go up (Q8 has 2-rank 1, but sectional 2-rank 2).  The definition of p-rank is on page 5 of Aschbacher's Finite Group Theory textbook. JackSchmidt (talk) 03:13, 13 May 2011 (UTC)
thread_title:  Definition of 2-rank? 

message: These groups can be seen as the basic building blocks of all finite groups in much the same way as the prime numbers are the basic building blocks of the natural numbers.
I find this sentence highly questionable and somewhat misleading. Ok for the first part, of course, but in much the same way... is not accurate: both prime factors of a given number and factors of a composition series of a given group are essentially unique, but the converse is false. A natural number is uniquely determined by its prime factors, but two "highly non-isomorphic groups" may share the same composition series! The comparison, as presented, could suggest that the extension problem is entirely trivial; on the contrary it is far from being trivial!--Paolo Lipparini (talk) 08:43, 14 April 2013 (UTC)
replies: text: I completely agree and came to this talk page to make exactly the same comment. Finite groups can be really quite complicated and the extension problem is very subtle. In some sense if you know about primes you know about what numbers there are. That is not true for finite groups. A question like "what groups are there of order n" for large n can be extremely difficult to answer. Some other terminology would be better. Perhaps "reminiscent"? 82.68.102.190 (talk) 17:15, 17 August 2013 (UTC)
thread_title: Comparison with factorization of natural numbers

message: I dont understand the article very well but I think there is a mistake. 
" Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof."
"1994 Gorenstein, Lyons, and Solomon begin publication of the revised classification"
Was he died when he started to publish the classification?  — Preceding unsigned comment added by 78.27.127.160 (talk) 19:17, 5 August 2013 (UTC)
replies: text: He died in 1992 but the publication is a write-up of earlier work to which he was a major contributor, so he is still listed as an author. 2601:648:8202:350:0:0:0:90B2 (talk) 01:55, 15 June 2022 (UTC)
thread_title: Mistake in Gorenstein's working years

message: I think someone expert might improve the article by writing why some facts listed in the timeline were useful in finding the proof. For example, there is a lonesome sentence "Conway introduces the Conway groups" but many readers would have no idea how that fact helps to prove the theorem. Also, a proper sketch of the whole theorem would be nice too. I know very well that it would be very challenging as the proof is so long.--78.27.127.160 (talk) 08:46, 16 September 2013 (UTC)
thread_title: Some thoughts how to improve the article

message: Substituted at 01:52, 5 May 2016 (UTC)
thread_title: Assessment comment

message: I don't know how to fix this, but only the first 6 published volumes of the second generation proof are being displayed in the second paragraph of that section, despite the fact that all 7 volumes are included in the "harv" template. It seems that the parameter year7=2018 is being ignored. Akriasas (talk) 12:55, 18 April 2018 (UTC)
thread_title:  Citation template for second-generation proof not working properly 

message: Hi, everyone. (I'm new to this.) The classification is just about the most impressive result in mathematics, so I think it deserves at least one nice visualization. Unfortunately, the only images of the classification that I was able to find online were confusing or misleading: they try to copy the form of the periodic table so much that it is no longer easy to see even how many families there are. Since I wanted to be constructive, I just made an image myself and I have now uploaded it. Please feel free to suggest improvements, which I will be happy to make. Also feel free to replace the image with anything better. Mathsies (talk) 14:31, 14 August 2021 (UTC)
thread_title:  Image summing up the classification 

message: I know that the absence of additional sporadic groups beyond the Monster is a big part of this proof, but could something be added to the article summarizing why there can't be any more?  It must be possible to give such an explanation (at least informally), since the result was expected long before it was proved.  Thanks. 2601:648:8202:350:0:0:0:90B2 (talk) 01:28, 15 June 2022 (UTC)
thread_title:  sporadic groups