有限單群分類

有限簡單群分類，亦被稱為龐大定理，在數學裡是個極巨大的工程。有關其的文章大多發表於1955年至1983年之間，目的在於將所有的有限簡單群都給分類清楚。這項工程總計約有100位作者在500篇期刊文章中寫下了上萬頁的文字。

分類

若結果是正確的話，分類表示每個有限簡單群都會是下列每類型的其中一種：

• 有著質數目的循環群
• 至少5階的交替群
• 「典型群」(於一有限體上之射特殊線性線、辛群、正交群或么正群)
• An exceptional or twisted group of Lie type (including the Tits group)
• 剩下26種被稱為散在群的其中之一(見下表)

此一定理在數學的許多分支都有著廣泛的應用，有關有限群的問題通常可以歸併至有關有限簡單群的問題上，再依此一分類即可將問題縮限至儘為各個例子的列舉上。

有時Tits group會被歸類為一種散在群(在此故而有27個散在群)，因為其不嚴格地為a group of Lie type。

散在群

散在群中的其中五種是在1860年代中由馬提厄(Mathieu)所發現的，而其他的21種則是在1965年至1975年之間被找出來的。有一些此類的群在它們被建構出來前曾被預測其會存在。大多數此類的群是以第一個預測出其存在之數學家來命名的。其完整的列表如下：

• 馬提厄群 M₁₁、M₁₂、M₂₂、M₂₃、M₂₄
• 詹柯群 J₁、J₂(HJ)、J₃(HJM)、J₄
• 康威群 Co₁、Co₂、Co₃
• 費歇爾群s Fi₂₂、Fi₂₃、Fi₂₄(Fi₂₄′)
• 希格曼-西姆斯群 HS
• 麥克勞林群 McL
• 赫爾得群 He(F₇)
• Rudvalis group Ru
• 鈴木散在群 Suz
• 歐南群 O'N
• 原田-諾頓群 HN(F₅)
• 里昂群 Ly
• 湯普森群 Th(F₃)
• 子怪獸群 B(F₂)
• 費歇爾-Griess 怪獸群 M(F₁)

Matrix representations over finite fields for all the sporadic groups have been computed, except for the Monster.

Of the 26 sporadic groups, 20 of them can be seen inside the Monster group as subgroups or quotients of subgroups. The 6 exceptions are J₁, J₃, J₄, O'N, Ru and Ly. These 6 groups are sometimes known as the pariahs.

So far, there has been little progress in providing a convincing unification for the sporadic groups.

Remaining skepticism on the proof

Some doubts remain on whether these articles provide a complete and correct proof, due to the sheer length and complexity of the published work and the fact that parts of the supposed proof remain unpublished. Jean-Pierre Serre is a notable skeptic of the claim of a proof. Such doubts were justified to an extent as gaps were later found and eventually fixed.

For over a decade, experts have known of a "serious gap" (according to Michael Aschbacher) in the (unpublished) classification of quasithin groups due to Geoff Mason. Gorenstein announced the classification of finite simple groups in 1983, based partly on the impression that the quasithin case was finished. Aschbacher filled this gap in the early 90s, also unpublished. Aschbacher and Steve Smith have published a different proof comprising two volumes of about 1300 pages.

二代分類

Because of the extreme length of the proof of the classification of finite simple groups, there has been a lot of work, called "revisionism", originally led by Daniel Gorenstein, in finding a simpler proof. This is the so-called second-generation classification proof.

Six volumes have been published as of 2005, and manuscripts exist for most of the rest. The two Aschbacher and Smith volumes were written to provide a proof for the quasithin case that would work with both the first- and second-generation proof. It is estimated that the new proof will be approximately 5,000 pages when complete. (It should be noted that the newer proofs are being written in a more generous style.)

Gorenstein and his collaborators have given several reasons why a simpler proof is possible. The most important is that the correct, final statement is now known. Techniques can be applied that will suffice for the actual groups. In contrast, during the original proof, nobody knew how many sporadic groups there were, and in fact some of the sporadic groups (for example, the Janko groups) were discovered in the process of trying to prove cases of the classification theorem. As a result, overly general techniques were applied.

Again, because the conclusion was unknown, and for a long time not even conceivable, the original proof consisted of many separate complete theorems, classifying important special cases. These proofs, in order to reach their own final statements, had to analyze numerous special cases. Often, most of the work was in these exceptions. As part of a larger, orchestrated proof, many of these special cases can be bypassed, to be handled when the most powerful assumptions can be applied. The price paid is that these original theorems, in the revised strategy, no longer have comparatively short proofs, but depend on the complete classification.

Nor were these separate theorems efficient regarding the subdivision of cases. Numerous target groups were identified multiple times as a result. The revised proof relies on a different subdivision of cases, eliminating these redundancies.

Finally, finite group theorists have more experience and new techniques.

參考文獻

• Michael Aschbacher, The Status of the Classification of the Finite Simple Groups, 2004年8月美國數學學會上的介紹
• Daniel Gorenstein, Richard Lyons, Ronald Solomon The Classification of the Finite Simple Groups (volume 1), AMS, 1994 (volume 2), AMS,
• Ron Solomon: On Finite Simple Groups and their Classification, Notices of the American Mathematical Society, February 1995
• Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.
• Orders of non abelian simple groups: includes a list of all non-abelian simple groups up to order 10,000,000,000.
• Atlas of Finite Group Representations: contains representations and other data for many finite simple groups, including the sporadic groups.