Talk:Classification of finite simple groups

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)

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)

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)

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)

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? -&#8472;yrop (talk) 18:51, Dec 12, 2004 (UTC)

Apparently there is: see Thompson groups. -&#8472;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)

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)

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

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)

History of finite simple groups
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)

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.-- Cala braxthis  08:31, 1 December 2007 (UTC)

I would also recommend a newly published book by Marcus du Sautoy, Finding Moonshine, Fourth Estate, 2008, ISBN 978-0-00-721461-7 which is suitable for the lay reader-- Cala braxthis  (talk) 11:03, 27 July 2008 (UTC)

Status of the second-generation classficication
<"As of 2005..."> Can anyone please update the status of the second-generation classification? Felix Hoffmann (talk) 06:52, 6 February 2010 (UTC)


 * 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)

Controversy?
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 |this 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)
 * 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)
 * 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)

Holder's result seems wrong
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)

I think the problem is imprecision in the statement. Of course, nonabelian finite simple group must be intended. Also, it is correct that the order of a finite nonabelian simple group has at least four prime factors (but the primes in the factorization need not be distinct), so this must be what is intended. The only case not covered by Burnside's p^a.q^b-theorem is then a potential simple group of order pqr where p,q,r are different primes. But there is no finite simple group G whose order has this form. This is not immediately obvious, but here's a reasonably elementary outline: let us suppose that p > q > r. If there is just one Sylow p-subgroup, then this is normal and G is not simple. The number of Sylow p-subgroups must divide the order of G, and be of the form tp+1 for some integer t. The only possibility is that there are qr Sylow p-subgroups. Hence there are qr(p-1) elements of order p in G. Let S be the set of elements of G which do not have order p -we don't know yet that this is a subgroup. Now S contains qr elements. Every element of order q in G must be in S, and S contains a subgroup of order r by Sylow's theorem for r (or by Cauchy's theorem will do in this case). Hence there are at most qr-r elements of order q in G. If there are qr-r elements of order q in G, then there is only one subgroup of order r in G, which must be normal. Otherwise, the number of Sylow q-subgroups of G is less than r. But the number of Sylow q-subgroups of G is a divisor of G, and r is the smallest divisor of the order of G other than 1, so there can be only one Sylow q-subgroup of G in this case, and then that is normal, so G is not simple. This argument has shown that if the order of G has the form pqr for three different primes pq and r, then at least one of the (non-identity) Sylow subgroups of G is normal, so G is not simple. There are more direct ways to prove this if a little more group theory is used. The alternating group of degree 5 does indeed show that it is possible for the order of a finite simple group to have the form p.q.r^2, for distinct primes p,q and rMessagetolove (talk) —Preceding undated comment added 23:28, 9 February 2011 (UTC).

Definition of 2-rank?
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)
 * 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)
 * Thanks. Is p-rank defined analogously?  Is there a standard textbook which contains this definition? Jowa fan (talk) 08:11, 10 May 2011 (UTC)


 * 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)

Comparison with factorization of natural numbers
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)


 * 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)

I like the periodic table analogy. The chemical formula doesn't tell you how the atoms go together. Isomers exist. If I had a "serious" source I'd add it to the article. --2607:FEA8:F8E1:F00:95F8:E103:5B90:4504 (talk) 04:39, 28 February 2024 (UTC)

Mistake in Gorenstein's working years
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)
 * 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)

Some thoughts how to improve the article
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)

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

The “four broad classes described below” are not clearly delineated below.
The first sentence of this article refers to “four broad classes described below”. Foolishly, I expected to see the 4 broad classes listed somewhere below in some obvious way. Eventually I concluded that they meant the three infinite classes, plus a fourth broad class that includes the 26 sporadic groups plus the Tits group, because the latter is “sometimes considered a 27th sporadic group”.

Did I guess right?

Could this be clarified? Cicada Cycle (talk) 04:54, 9 November 2017 (UTC)


 * Well the statement of the theory sections has


 * which seems pretty clear. --Salix alba (talk): 06:30, 9 November 2017 (UTC)


 * The plain fact is that the "four" broad classes are slightly asymmetrical. The first three infinite classes is clear enough–it is that fourth "class" that causes confusion.  First of all, it is a finite class–there are just 27 of them.  Then there is the annoying Tits group that sticks out like a sore thumb.  That is just the way it is.  In any case, upon careful reading, the statement of the theorem is perfectly clear.--Toploftical (talk) 03:15, 10 November 2017 (UTC)

Citation template for second-generation proof not working properly
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)

Image summing up the classification
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)

sporadic groups
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)