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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
673:
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 r
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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.
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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.
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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 : -->
260:; 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.
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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
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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
422:
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
1162:
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
991:
The first sentence of this article refers to âfour broad classes described belowâ. 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
233:
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
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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
669:
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
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The plain fact is that the "four" broad classes are slightly asymmetrical. The first three infinite classes is clear enoughâit is that fourth "class" that causes confusion. First of all, it is a finite classâthere are just 27 of them. Then there is the annoying Tits group that sticks out like a
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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
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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
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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
412:
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
654:
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).
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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
248:
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.
283:
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.
630:(GTM 251) that "With the publication of the two volumes by Aschbacher and Smith in 2004 we can reasonably regard the proof of the Classification Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete." Those two citations "" refer to the two-volume book
442:
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
225:
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. --
788:-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
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E. Mathieu, 'Mémoire sur l'étude des functions des plusieurs quantités, sur le maniÚre de les former et sur les sustitutions qui les laissent invariables', J. Math. Pures Appl. (Liouville) (2) 6 (1861),
886:" 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"
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23:43, 14 April 2007 (UTC) Needs more history of discovery of sporadic groups and classification, but already close to B+ class. It's definitely worthwhile improving it to A class or GA.
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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.
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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.
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natural enough - both relate to peculiarities of 24-dimensional spaces. I suppose the
Monster and the Griess algebra represent a similar expansion. 12:57 15 Mar 2006
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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? --
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472:) for anyone seeking a historical overview of the discovery of the 26 exceptional finite simple groups and the initiation of the classification programme.--
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846:"what groups are there of order n" for large n can be extremely difficult to answer. Some other terminology would be better. Perhaps "reminiscent"?
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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.
864:. 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. --
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1123:, despite the fact that all 7 volumes are included in the "harv" template. It seems that the parameter year7=2018 is being ignored.
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The term "2-rank" is used frequently in this article, but not defined. Does anyone know precisely what it means? Is it related to
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appeared in the 1861 paper and the 3 large groups in the 1873 paper. They are cited in Dixon & Mortimer's "Permutation groups".
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932:
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.--
961:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
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sore thumb. That is just the way it is. In any case, upon careful reading, the statement of the theorem is perfectly clear.--
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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
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I don't know how to fix this, but only the first 6 published volumes of the second generation proof are being displayed in
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There has been some controversy as to whether the resulting proof is complete and correct, given its length and complexity
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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.
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on p. 108: "It is quite impossible for a layman to judge whether a complete proof of the theorem currently exists."
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Is there any such thing as an infinite simple group, and if not, then would it make sense to merge this page into
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There are many infinite simple groups. Having this page on the finite simple case is more than justified.
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I find this sentence highly questionable and somewhat misleading. Ok for the first part, of course, but
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appropriate to add to the article, but just FYI there are still some significant holes in the proof.
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groups plus the Tits group, because the latter is âsometimes considered a 27th sporadic groupâ.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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Answer: Mathieu published papers in 1861 and 1873. I have not seen them, but I understand that M
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stands for Hall-Janko should be left to the group's own page. Each group has its own story.--
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The 2-rank of a group G is the maximum number of generators of an Abelian 2-subgroup of G.
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750:-rank defined analogously? Is there a standard textbook which contains this definition?
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No, the classic simple groups are certain quotients of linear groups over finite fields.
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Last edited at 09:05, 24 May 2007 (UTC). Substituted at 01:52, 5 May 2016 (UTC)
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article; so I don't really see that the topic can be done justice, right now.
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Can anyone please update the status of the second-generation classification?
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The âfour broad classes described belowâ are not clearly delineated below.
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I dont understand the article very well but I think there is a mistake.
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didn't understand it well enough to write a meaningful/accurate summary.
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could someone give a "basic idea" of the original classification program?
555:, with an indication that it still had not changed as of February 2014.
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There's presumably some additional condition needed. What is missing?
776:-rank of a finite group is the largest rank of an elementary abelian
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Anyone want to write a bit about each of the sporadic groups? :-) --
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is entirely trivial; on the contrary it is far from being trivial!--
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Citation template for second-generation proof not working properly
792:-rank is on page 5 of Aschbacher's Finite Group Theory textbook.
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In the case of groups of Lie type, we are still waiting for a
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Was he died when he started to publish the classification?
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a member of one of three infinite classes of such, namely:
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I'm sure there is much more to say, but I'm not an expert!
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Meanwhile, Wilson notes in the preface of his 2009 book
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1072:(which is sometimes considered a 27th sporadic group).
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Talk:Classification of finite simple groups/Comments
112:, a collaborative effort to improve the coverage of
784:-rank is the largest rank of an elementary abelian
826:comparison, as presented, could suggest that the
492:I would also recommend a newly published book by
812:Comparison with factorization of natural numbers
1207:Knowledge level-4 vital articles in Mathematics
957:, and are posted here for posterity. Following
531:Status of the second-generation classficication
465:by Mark Ronan (Oxford University Press, 2006,
1013:Well the statement of the theory sections has
951:The comment(s) below were originally left at
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551:The status as of November 2012 is addressed
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927:Some thoughts how to improve the article
461:As a humble lay reader, I can recommend
1197:Knowledge vital articles in Mathematics
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866:2607:FEA8:F8E1:F00:95F8:E103:5B90:4504
632:The Classification of Quasithin Groups
507:which is suitable for the lay reader--
1212:B-Class vital articles in Mathematics
879:Mistake in Gorenstein's working years
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1121:the second paragraph of that section
106:This article is within the scope of
1138:Image summing up the classification
49:It is of interest to the following
861:I like the periodic table analogy
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1222:Top-priority mathematics articles
959:several discussions in past years
126:Knowledge:WikiProject Mathematics
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129:Template:WikiProject Mathematics
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146:This article has been rated as
1202:B-Class level-4 vital articles
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1061:one of 26 groups called the "
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874:04:39, 28 February 2024 (UTC)
418:17:54, 22 February 2007 (UTC)
335:Scott Tillinghast, Houston TX
307:Scott Tillinghast, Houston TX
194:Chas, you're thinking of the
120:and see a list of open tasks.
1217:B-Class mathematics articles
1165:2601:648:8202:350:0:0:0:90B2
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1078:which seems pretty clear. --
1008:04:54, 9 November 2017 (UTC)
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665:21:01, 9 February 2011 (UTC)
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650:Holder's result seems wrong
453:07:50, 14 August 2008 (UTC)
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780:-subgroup. The sectional
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998:Could this be clarified?
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1163:it was proved. Thanks.
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628:The Finite Simple Groups
535:<"As of 2005...": -->
463:Symmetry and the Monster
292:21:35, 16 May 2004 (UTC)
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152:project's priority scale
823:in much the same way...
608:wrote in his 1999 book
585:However, it seems that
500:, Fourth Estate, 2008,
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343:Infinite simple groups?
222:15:34 4 Jul 2003 (UTC)
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169:Classical simple groups
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279:Mathieu group dates
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