• 沒有找到結果。

We present some highlights of the 110-year project to classify the finite simple groups


Academic year: 2022

Share "We present some highlights of the 110-year project to classify the finite simple groups"


加載中.... (立即查看全文)



Volume 38, Number 3, Pages 315–352 S 0273-0979(01)00909-0

Article electronically published on March 27, 2001



Abstract. We present some highlights of the 110-year project to classify the finite simple groups.

1. The beginnings

“Es w¨are von dem gr¨ossten Interesse, wenn eine Uebersicht der s¨ammtlichen ein- fachen Gruppen von einer endlichen Zahl von Operationen gegeben werden k¨onnte.”

[“It would be of the greatest interest if it were possible to give an overview of the entire collection of finite simple groups.”] So begins an article by Otto H¨older in Mathematische Annalen in 1892 [Ho]. Insofar as it is possible to give the birthyear of the program to classify the finite simple groups, this would be it. The first paper classifying an infinite family of finite simple groups, starting from a hypothesis on the structure of certain proper subgroups, was published by Burnside in 1899 [Bu2].

As the final paper (the classification of quasithin simple groups of even characteris- tic by Aschbacher and S. D. Smith) in the first proof of the Classification Theorem for the Finite Simple Groups (henceforth to be called simply the Classification) will probably be published in the year 2001 or 2002, the classification endeavor comes very close to spanning precisely the 20th century.

Of course there were some important pre-natal events. Galois introduced the concept of a normal subgroup in 1832, and Camille Jordan in the preface to his Trait´e des substitutions et des ´equations algebriques in 1870 [J1] flagged Galois’

distinction between groupes simples and groupes compos´ees as the most important dichotomy in the theory of permutation groups. Moreover, in the Trait´e, Jordan began to build a database of finite simple groups – the alternating groups of degree at least 5 and most of the classical projective linear groups over fields of prime cardinality. Finally, in 1872, Ludwig Sylow published his famous theorems on subgroups of prime power order [Sy].

Nevertheless H¨older’s paper is a landmark. H¨older threw down a gauntlet which was rapidly taken up by Frank Cole, who in 1892 [Co1] determined all simple groups of orders up to 500 (except for some uncertainties related to 360 and 432) and in 1893 [Co2] extended this up to 660, discovering in the process a new simple group SL(2, 8). By the dawn of the 20th century Miller and Ling (1900) [ML] had pushed this frontier out to 2001. These results were achieved with the only available tools – Sylow’s Theorems and the Pigeonhole Principle. Needless to say, the arsenal of

Received by the editors September 18, 2000, and in revised form December 15, 2000.

2000 Mathematics Subject Classification. Primary 20D05.

Research partially supported by an NSF grant.

2001 American Mathematical Societyc 315


weapons needed to be enlarged and the strategy of proceeding one integer at a time needed to be abandoned if any serious progress was to be made.

An alternate strategy was already implicit in a lemma of H¨older (1892) [Ho]

proving that a simple group whose order is a product of at most three prime numbers must be cyclic of prime order. The idea that the structure of a finite group G depends more on the shape of the prime factorization of |G| than on the actual nature of the prime factors was explicitly stated by Cole and Glover (1893) [CG]

in their critique of Cayley and Kempe: “It is however a defect of their method of classification that it proceeds simply according to the order and not the type of the groups. Thus the groups of order pq where p and q are prime numbers are all of one of two types, the orders 10, 14, 15, . . . presenting no greater complexity than the order 6.” This vein was explored further by Burnside and Frobenius, who both established by 1895 that the only nonabelian simple groups whose order is a product of at most five prime factors are P SL(2, p) for p ∈ {5, 7, 11, 13}. As remarked by Peter Neumann, it is an open question analogous to the Twin Primes Conjecture whether there are infinitely many primes p for which |P SL(2, p)| is a product of exactly six prime factors. Thus again this strategy rapidly encounters difficult and irrelevant obstacles. Its last hurrah was Burnside’s proof in 1900 [Bu3]

that if G is a nonabelian simple group of odd order, then|G| must be a product of at least seven prime numbers.

In a similar but more fruitful direction, Frobenius [Fr1] proved in 1893 that a simple group of squarefree order must be cyclic of prime order. Burnside extended this in 1895 [Bu1] to the following suggestive result:

Theorem. If p is the smallest prime divisor of |G| and if G has a cyclic Sylow p-subgroup P , then G = KP , where K is a normal subgroup of order prime to p.

In particular if G is simple, then|G| = p.

This points to the importance of the smallest prime divisor p of|G| and to the significance of the structure of a Sylow p-subgroup, not simply its cardinality.

Thus by 1895 the strategies were becoming more subtle, but the techniques had not advanced beyond Sylow and virtuoso counting arguments. All of this began to change on April 6, 1896, when Dedekind wrote his famous letter to Frobenius inviting him to consider the problem of factoring the group determinant of a finite nonabelian group. This problem, which vastly generalizes the problem of factoring the determinant of circulant matrices, is roughly equivalent to the decomposition of the regular representation of a finite group into its irreducible constituents, or the complex group algebra as a sum of simple two-sided ideals. The solution by Frobenius later that year signalled the birth of the theory of group characters, which he rapidly developed to the point where he could give a recursive algorithm for the computation of the group characters of the symmetric groups Sn for all n. Nevertheless it was not immediately clear whether this new theory of group characters was of any use for the structure theory of finite groups.

This question was resolved in the affirmative by Burnside in 1900 [Bu3] when he used the new theory to prove that a transitive permutation group of prime degree p either must be 2-transitive or must have a normal Sylow p-subgroup of order p.

Since a 2-transitive group G of degree p must have|G| divisible by p(p−1), G must in particular either be of even order or be solvable. Using this, Burnside was able to show that if G is a nonabelian simple group of odd order, then|G| > 40000, |G|

must have at least seven prime factors, and G can have no proper subgroup of index


less than 101. This prompted his famous observation in 1911 [Bu5]: “The contrast that these results shew between groups of odd and even order suggests inevitably that [nonabelian] simple groups of odd order do not exist.”

The less-than-friendly rivalry between Frobenius and Burnside produced two further spectacular applications of group characters. Both represented the culmi- nation of a sequence of partial results by both researchers. First came Frobenius’

Theorem. We call a permutation group G on a set X regular if, for any two points x, y∈ X, there exists a unique g ∈ G with g(x) = y. Every group acts regularly on itself via the regular representation. Thus regular action puts no restriction on the structure of G. On the other hand Frobenius [Fr2] proved the following result.

Frobenius’ Theorem. Let G be a permutation group which acts transitively but not regularly on a finite set X. Suppose that no nonidentity permutation in G fixes more than one point. Then G = KGx, where K is a regular normal subgroup of G and Gx is the stabilizer of the point x. In particular G is not a simple group.

There is an equivalent theorem in the context of abstract groups. If H is a subgroup of G, we let Hg= g−1Hg.

Theorem. Let G be a finite group with a proper subgroup H such that H∩Hg= 1 for all g∈ G − H. Then G has a proper normal subgroup K such that G = KH and K∩ H = 1.

Groups G satisfying the hypotheses of either version of Frobenius’ Theorem are called Frobenius groups, and the subgroups H and K are called a Frobenius complement (unique only up to G-conjugacy) and the Frobenius kernel respectively.

In the proof Frobenius used his Reciprocity Theorem. As clarified later by Brauer and Suzuki, the key is the decomposition of certain virtual characters induced from H to G, yielding characters of G which restrict irreducibly to H. Then K may be recognized as the kernel of a suitable character of G.

The final triumph of this era of the Classification was Burnside’s proof in 1904 [Bu4] of the paqb Theorem, using the arithmetic of group characters.

Burnside’s paqb Theorem. Let G be a finite group such that|G| = paqbfor some primes p and q. Then G is a solvable group.

Burnside had great hopes for further applications of character theory to the study of finite groups, in particular to the proof of the nonexistence of nonabelian simple groups of odd order. This would have to wait half a century. But before moving on, it is worthwhile to mention the early developments in the American school.

A vigorous school of mathematics was nurtured at the University of Chicago in the 1890’s by E.H. Moore, and finite group theory was one of its major interests.

Moore proved that any complex linear representation of a finite group is a unitary representation, and using this Maschke proved that every complex linear represen- tation is completely reducible. Moore’s student, L.E. Dickson, extended Jordan’s database of simple groups to include all the classical projective groups over finite fields in his book Linear Groups (1900) [D1]. Dickson was well aware of the analogy between his work and the recent monumental results of Killing and Cartan classi- fying the simple continuous groups of Lie, and shortly thereafter (1901,1903) [D2], [D3] he succeeded in constructing analogues of the Lie group G2 over finite fields F and establishing their simplicity when|F | > 2. He also studied finite analogues of E6 but did not prove their simplicity.


2. Theory building

Dickson declared finite group theory to be dead in the 1920’s. Indeed there was something of a hiatus from World War I to the 1930’s. Nevertheless there were soon important new developments. First came the work of Philip Hall, Burnside’s intellectual heir. Hall acknowledged this genealogy in 1942 at the time of his election to the Royal Society (quoted in [Ro]):

“The aim of my researches has been to a very considerable extent that of extend- ing and completing in certain directions the work of Burnside. I asked Burnside’s advice on topics in group theory which would be worth investigation and received a postcard in reply containing valuable suggestions as to worthwhile problems. This was in 1927 and shortly afterwards Burnside died. I never met him, but he has been the greatest influence on my ways of thinking.”

Hall undertook in 1932 [H2] “the first stages of an attempt to construct a sys- tematic general theory of groups of prime-power order,” justifying this project with the sentence: “It is widely recognised, I believe, that the astonishing multiplicity and variety of these groups is one of the main difficulties which beset the advance of finite-group-theory.” Indeed one feature of the later history of the Classification is an assiduous and largely successful effort to avoid having to study any p-groups beyond those masterfully analyzed by Philip Hall.

Even more important for the future theory of simple groups was the series of papers Hall published in 1928 [H1] and 1937 [H3], [H4] on finite solvable groups.

In the first he establishes generalizations of Sylow’s Theorems for finite solvable groups; namely if G is a finite solvable group of order mn with m and n coprime, then G possesses a subgroup H of order m, any two such subgroups are conjugate and any subgroup of order dividing m is contained in a conjugate of H. Such a subgroup H is now called a Hall subgroup or Hall π-subgroup where π is the set of prime divisors of m. More striking were Hall’s results of 1937 showing that the existence of Hall π-subgroups is a characteristic property of finite solvable groups.

Thus, although a nonsolvable group G may have Hall π-subgroups for certain sets of primes, e.g. {2, 3} for A5 and P SL(2, 7), the following theorem holds.

P. Hall’s Theorem. G is a finite solvable group if and only if, for each expression

|G| = mn with m prime to n, G contains at least one subgroup of order m and at least one subgroup of order n.

This theorem extends Burnside’s paqb Theorem in a very suggestive fashion, re- lating the existence of nontrivial normal subgroups in a group G to the factorization of G as a product of permutable subgroups G = M N = N M (of coprime orders).

The connection between solvability and factorizations was explored further in the 1950’s by Wielandt, Hall, Kegel and others. Finally this line of thought culmi- nated in the factorization theorems of Thompson, which are among the key ideas of the classification proof. Hall’s elegant and elementary proof relies on Burnside’s theorem as the base case. Coming full circle, Thompson’s factorization theorems and related ideas of Bender would finally lead in 1973 to a character-free proof of Burnside’s Theorem by Goldschmidt [Go1] and Matsuyama [Ma].

In Germany the 1920’s witnessed intense activity on the structure theory of many algebraic objects – fields, division algebras, rings with ACC or DCC, etc. Interest in finite groups led to important papers in the late 1930’s by Fitting, Wielandt and


Zassenhaus in particular. The climate of thought is so well captured by Zassen- haus in the preface to his 1937 book Lehrbuch der Gruppentheorie (translated into English and published by Chelsea in 1949 [Z3]) that I quote it at length:

“Investigations published within the last fifteen years have greatly deepened our knowledge of groups and have given wide scope to group-theoretic methods. As a result, what were isolated and separate insights before, now begin to fit into a unified, if not yet final, pattern....

“It was a course of E. Artin, given in Hamburg during the Winter Semester of 1933 and the Spring Semester of 1934, which started me on an intensive study of group theory. In this course, the problems of the theory of finite groups were transformed into problems of general mathematical interest. While any question concerning a single object (e.g., finite group) may be answered in a finite number of steps, it is the goal of research to divide the infinity of objects under investigation into classes of types with similar structure.

“The idea of O. H¨older for solving this problem was later made a general principle of investigation in algebra by E. N¨other. We are referring to the consistent appli- cation of the concept of homomorphic mapping. With such mappings one views the objects, so to speak, through the wrong end of a telescope. These mappings, applied to finite groups, give rise to the concepts of normal subgroups and of factor groups. Repeated application of the process of diminution yields the composition series, whose factor groups are the finite simple groups. These are, accordingly, the bricks of which every finite group is built. How to build is indicated – in principle at least – by Schreier’s extension theory. The Jordan-H¨older-Schreier theorem tells us that the type and the number of bricks is independent of the diminution process.

The determination of all finite simple groups is still the main unsolved problem.”

(“Als ungel¨ostes Hauptproblem verbleibt die Bestimmung aller endlichen einfachen Gruppen.”)

In contrast to earlier texts by Burnside [Bu5], Miller, Blichfeldt and Dickson [MBD] (1916) and Speiser [Sp] (1927), Zassenhaus’ work almost completely omits the theories of permutation groups and of group characters, focussing singlemind- edly on the architectural structure of groups in terms of normal subgroups and factor groups. In addition to his improved treatment of the Jordan-H¨older-Schreier Theorem, the principal additions are an exposition and amplification of the theory of group extensions developed by Schur and Schreier. Most notable in this context is the following theorem.

Schur-Zassenhaus Theorem. Let G be a finite group with a normal subgroup N such that |N| and |G/N| are coprime. Then the extension G splits over N; i.e.

there is a subgroup H of G with G = N H and N ∩ H = 1. Moreover if either N or G/N is a solvable group, then all complements to N in G are G-conjugate.

Zassenhaus notes that the conjugacy of complements had been conjectured to hold without any restriction of solvability and observes that the problem had been reduced by Witt to the case where N is simple and CG(N ) = 1; i.e. G is a subgroup of Aut(N ). He then observes that the desired general theorem would follow from the proof of either of the following conjectures:

Odd Order Conjecture (Miller, Burnside). Every finite group of odd order is solvable.


Schreier Conjecture. If N is a nonabelian finite simple group, then Aut(N )/N is a solvable group.

Both conjectures turned out to be true, but very deep. Indeed the only known proof of the Schreier Conjecture is as a corollary of the Classification Theorem.

The proof of the Odd Order Conjecture by Feit and Thompson [FT] finally yielded the unrestricted Schur-Zassenhaus Theorem, for which there is still no known ele- mentary proof.

In the same architectural spirit was the paper “Beitr¨age zur Theorie der endlichen Gruppen” by Fitting [F], edited and published posthumously by Zassenhaus in 1938.

In this paper Fitting took a somewhat different approach to the group extension problem and most notably focussed attention on what came to be called the Fit- ting subgroup F (G) of a finite group, G, namely the join of all normal nilpotent subgroups of G. He showed that F (G) is itself nilpotent, hence the unique largest normal nilpotent subgroup of G. Moreover for solvable groups G he singled out the following fundamental property.

Fitting’s Theorem. Let G be a finite solvable group. Then CG(F (G))≤ F (G).

Since F (G) is a normal (indeed, characteristic) subgroup of G, the conjugation action of G on the elements of F (G) defines a homomorphism of G into Aut(F (G)) whose kernel is CG(F (G)). Thus an equivalent formulation of Fitting’s Theorem is the following result.

Theorem. Let G be a finite solvable group. Then G/Z(F (G)) is isomorphic to a subgroup of Aut(F (G)).

The corresponding assertion is emphatically false for general finite groups. In- deed if G is a nonabelian simple group, then F (G) = 1. The search for a good substitute for F (G) when G is non-solvable would occupy the attention of Goren- stein, Walter and Bender in the late 1960’s, but even before this, Fitting’s Theorem would play an important role in the thinking of Philip Hall in the 1950’s.

There were numerous other notable developments in the 1930’s. Gr¨un [Gu] and Wielandt [Wi] refined and extended Burnside’s transfer homomorphism. Given a Sylow p-subgroup P of a finite group G, the idea of the transfer is to use the nat- ural surjection P → P/[P, P ] to define a homomorphism VG→P/[P,P ] which under suitable hypotheses may be shown to be nontrivial. The goal is to detect a normal subgroup N of G such that the quotient group G/N is an abelian p-group. For example if G is a finite group of order n with a cyclic Sylow 2-subgroup gener- ated by the element t, then the regular representation of G represents t as an odd permutation on n letters. In other words the kernel N of the induced homomor- phism G → Sn/An is a normal subgroup of G with G/N cyclic of order 2. This argument was known to Frobenius and Burnside, and Burnside realized that it could be extended to produce cyclic quotients of order greater than 2 by replacing permutation representations by monomial representations, i.e. by considering ho- momorphism G→ GL(m, C)/SL(m, C). This was refined around the 1930’s into the modern definition of the transfer homomorphism, which served as a tool in abelian class field theory as well as finite group theory.

This is a good moment to digress slightly to explain the concept of local group theory which began to emerge at this point. As defined later by Alperin, a local (or p-local) subgroup of a finite group G is the normalizer in G of a nonidentity p- subgroup of G for some prime p. If G is a nonabelian simple group, then every local


subgroup N is a proper subgroup of G. Indeed the local subgroups afford the largest proper subgroups of G whose existence can be predicted a priori from knowledge of |G| alone. If |G| = pag0 with (p, g0) = 1, then Sylow’s theorems guarantee the existence of subgroups of order pb for 1≤ b ≤ a and the normalizers of these subgroups are the local subgroups of G. The main challenge confronted in the Classification Project was: Can we use the “global” hypothesis of the simplicity of G to limit (severely) the possibilities for the p-local data of G, i.e. the isomorphism types and embeddings of the p-local subgroups of G? Or contrapositively, can we show that most sets of hypothetical p-local data for G are incompatible with the simplicity of G? If this were not possible, then any inductive classification enterprise would have necessarily foundered on the shoals of the unthinkably large number of possible sets of p-local data.

For example, in the context of the transfer homomorphism, wishful thinking might suggest the following claim: If G has no nontrivial abelian p-quotient, then some p-local subgroup N of G has no nontrivial abelian p-quotient. Unfortunately this statement is false, as is easily seen by inspection of the alternating group A6, which has no nontrivial 2-quotient. On the other hand every 2-local subgroup is isomorphic either to a dihedral group of order 8 or to the symmetric group S4, both of which have nontrivial 2-quotients. On the bright side, however, Burnside proved that the claim is true when G has an abelian Sylow p-subgroup P , taking N = NG(P ), the normalizer in G of P . Wielandt [Wi] extended this considerably (for odd p) to the case when P is a regular p-group in the sense of Philip Hall. In particular this covers the cases when P has exponent p and when P has nilpotence class less than p (i.e. the p-fold commutator [x1, x2, . . . , xp] = 1 for all xi∈ P ).

This is about as far as you can go while focussing on NG(P ). The search for suitable alternative p-local subgroups led Gr¨un and Wielandt to the concept of a weakly closed subgroup, which would later loom large in the thinking of Thompson.

Definition. Let H be a subgroup of the group G. A subgroup W of H is weakly closed in H (with respect to G) if Wg≤ H implies Wg= W for all g∈ G; i.e. W is the unique member of its G-conjugacy class which is contained in H.

Gr¨un [Gu] proved that if the center Z(P ) is a weakly closed subgroup of P , then G has an abelian p-quotient if and only if NG(Z(P )) does. This turned out to be extendible, in a spirit similar to Wielandt’s extension of Burnside’s theorem, to what might be dubbed regularly embedded weakly closed subgroups of P . The best result in this vein was achieved in the 1970’s by Yoshida [Y].

The most general theorem, again of later vintage, is Alperin’s Fusion Theorem (1967) [Al], which implies that the existence of abelian p-quotients is always de- termined p-locally, i.e. by examination of the full set of p-local data for G. Indeed it provides much sharper information. Also in the late 1960’s, Glauberman [Gl3]

defined nontrivial subgroups K(P ) and K(P ) such that the normalizer of either of them detects nontrivial p-quotients of G whenever p≥ 5. Ironically, however, by this point the role of the transfer homomorphism in the Classification was rapidly waning. Moreover, as attention focussed, after the Odd Order Theorem, on the 2- local structure of G, what little interest transfer held focussed mostly on the prime 2, and here the most effective tool proved to be the Thompson Transfer Lemma, which is merely an elementary refinement of the old permutation group argument of Frobenius and Burnside.


Yet another major development in the 1930’s was Zassenhaus’ extension of the work of Jordan and Frobenius on transitive permutation groups. Jordan had clas- sified all sharply k-transitive permutation groups for k = 2 and for k ≥ 4 [J2].

Sharply 2-transitive permutation groups of a finite set Ω satisfy the hypotheses of Frobenius’ Theorem and thus have the structure G = KGα where K is a regular normal subgroup of G. As K− {1} is permutation-isomorphic to Ω − {α} as Gα- set, K must be an elementary abelian p-group for some prime p with |Ω| = pn, n≥ 1. Burnside attempted a description of the structure of Gα, but the first cor- rect treatment was given by Zassenhaus in 1936 [Z2]. At the same time, Zassenhaus classified all finite sharply 3-transitive permutation groups G [Z1]. In such a group G, the point stabilizer Gα is a sharply 2-transitive group. Zassenhaus initiated the investigation of the larger class of finite 2-transitive permutation groups in which a point stabilizer is a Frobenius group. Such groups came to be known as Zassen- haus groups, and their classification in the late 1950’s and early 1960’s was a major project involving the work of Thompson, Feit, Ito, G. Higman and Suzuki. The most dramatic moment was Suzuki’s discovery [Su3] of a new infinite family of finite nonabelian simple Zassenhaus groups, all of order prime to 3. These groups came to be known as the Suzuki groups, Sz(22n+1), n ≥ 1. Suzuki’s work culminated in 1963 with the following theorem [Su4], [Su5] which extends his classification of Zassenhaus groups of odd degree.

Suzuki’s Theorem. Let G be a finite simple 2-transitive permutation group on a set Ω with|Ω| odd. Suppose that for α, β ∈ Ω, |Gαβ| is odd and Gαβ has a normal complement Q in Gα such that Q is regular on Ω− {α}. Then G ∼= P SL(2, 2n), Sz(22n+1) or P SU (3, 2n).

Probably the deepest group-theoretic work in the 1930’s was the investigation of the modular representations of finite groups, primarily by Richard Brauer, who had left Germany in 1933 and settled in Toronto in 1935 after a year as Weyl’s assistant at the IAS. In Brauer’s modular theory, the local/global principle reaches a higher level of refinement, as Brauer moves back and forth both between the characteristic 0 and the characteristic p representations of a group G and also between the representations of G and the representations of the p-local subgroups of G. His early work culminated in his paper on the p-modular representations of finite groups G such that|G| is divisible by p but not by p2, presented to the A.M.S. in 1939 [Br1]. As reported by Feit [Fe], one of Brauer’s “motives in studying modular representations was the hope of characterizing certain classical groups over finite fields.” Indeed in [Br1] he notes applications such as the uniqueness of simple groups of order 5616 and 6048. The fundamental principles of his analysis later evolved into his general Main Theorems. The calculation of so-called decomposition matrices, which underlie the detailed results he achieved for Sylow groups of prime order, soon founders however on the rocks of wild representation type. Nevertheless Brauer was later able to push through the analysis for certain Sylow 2-subgroups of 2-rank 2, which proved to be just what was needed to obtain group order formulas for simple groups with dihedral, semidihedral and “wreathed” Sylow 2-subgroups.

Alas in general his theory was of much less applicability to the Classification than he had hoped.


3. The Classification begins in earnest

After the trauma of World War II, the simple group problem recaptured the attention of leading algebraists. “Das interessanteste Problem scheint mir die Auf- suchung aller endlichen einfachen Gruppen zu sein.” [“The most interesting prob- lem seems to me to be the classification of all finite simple groups.”] So spoke Zassenhaus in his Antrittsvorlesung at the University of Hamburg in 1947 (quoted in [Be5]). The interest in the problem was clear. The best strategy for its solution was not at all clear. Inspired by the work of Killing and E. Cartan, Zassenhaus hoped to linearize the problem by identifying all simple groups as groups of auto- morphisms of some linear structure, perhaps a finite Lie algebra. This approach gained credibility when Chevalley [Ch] found a uniform method to construct fi- nite analogues of the simple complex Lie groups. Supplemented with variations by Steinberg [St] and Ree [Re1], [Re2], this furnished a Lie-theoretic context for all of the known finite simple groups except for the alternating groups and the five Mathieu groups. However, although this yielded a rich harvest of new finite simple groups and a unified context for the study of their subgroups and representations, it did not immediately suggest a strategy for their classification. Indeed, no one has yet found an a priori method of associating to a finite simple group G an algebra of the “correct” dimension, roughly log(|G|).

Meanwhile around 1950 several mathematicians independently began parallel investigations which were to prove more immediately fruitful for the classification endeavor. The groups P SL(2, q) are the most elementary of the nonabelian simple groups, and their subgroups have been well-understood since the time of E.H. Moore and Dickson. When q is a power of 2, the centralizer of every nonidentity element is abelian and the group is partitioned as a union of abelian (Sylow 2-)subgroups of order q and cyclic (Hall) subgroups of order q−1 and q+1, any two intersecting only in the identity. When q≡  (mod 4) (with  = ±1), the centralizer of every element of order greater than 2 is abelian and the group is partitioned as a union of abelian (Sylow p-)subgroups of order q and cyclic subgroups of order q−  and q+2 , again any two intersecting only in the identity. On the other hand, if G is a finite group in which the centralizer of every non-identity element is abelian, then it is not hard to see that the commuting graph of nonidentity elements of G is a union of disjoint cliques, corresponding to a partition of G as a union of maximal abelian subgroups.

Furthermore the normalizer of any such maximal abelian subgroup A is a Frobenius group with Frobenius kernel A. When a group G has such a subgroup structure, the situation is especially favorable for the analysis of induced virtual characters (first studied by Brauer) in the spirit of Frobenius’ Theorem. This seems to have been recognized independently around 1950 by Brauer, Suzuki and G.E. Wall, who began parallel investigations.

In 1948 Brauer accepted a chair at the University of Michigan and shortly there- after, together with his student K.A. Fowler, began the investigation of CA-groups of even order, i.e. groups of even order in which the centralizer of every nonidentity element is abelian. Around the same time, Wall in Manchester began similar re- search at the suggestion of Graham Higman. (There were some antecedents in the work of Szekeres (1949)[Sz] and Redei (1950)[Rd].) Meanwhile Suzuki in Japan dis- covered a characterization of P GL(2, q), q odd, in terms of partitions [Su1], which attracted the attention of Baer, who invited Suzuki to join him at the University of Illinois in 1951. The following summer Suzuki participated in Brauer’s summer


seminar in Michigan, which was also attended by graduate students including Wal- ter Feit and John H. Walter. By 1953, Brauer, Suzuki and Wall had each arrived at characterization theorems for P SL(2, q) beginning from local data, one version of which would be published several years later as the Brauer-Suzuki-Wall The- orem [BSW]. Surprisingly, the easy case of this theorem together with some of the key non-character-theoretic ideas for its proof had been discovered and pub- lished in 1899 by Burnside [Bu2], a result of which deserves recognition as the first classification theorem for finite simple groups:

Theorem (Burnside, 1899). Let G be a finite nonabelian simple group of even order. Suppose that every element of G is either an involution (i.e. an element of order 2) or is of odd order. Then G ∼= SL(2, 2n) for some integer n≥ 2.

Note that Burnside’s hypothesis can be reformulated in the CA-spirit as follows:

Suppose that G is a group of even order in which the centralizer of every invo- lution of G is an abelian group of exponent 2.

Burnside’s beautiful paper dwelt in undeserved obscurity until it was rediscovered by Walter Feit around 1970 to the great surprise of Brauer. It is an example of an important paper which had no impact on the history of the field.

In 1952 Brauer moved to Harvard and in 1954 he addressed the International Congress of Mathematicians, beginning with the following words:

“The theory of groups of finite order has been rather in a state of stagnation in recent years. This has certainly not been due to a lack of unsolved problems. As in the theory of numbers, it is easier to ask questions in the theory of groups than to answer them. If I present here some investigations on groups of finite order, it is with the hope of raising new interest in the field.”

Brauer focussed primarily on groups of even order and announced, in addition to the Brauer-Suzuki-Wall Theorem on P SL(2, q), the Brauer-Fowler bound on the order of a finite simple group of even order, given the order of one of its involution centralizers, and a characterization of P SL(3, q) and M11 via the centralizer of an involution. Because of its generality, the Brauer-Fowler Theorem [BF] had a particularly great psychological impact.

Brauer-Fowler Theorem. Let G be a finite simple group of even order containing an involution t. If|CG(t)| = c, then |G| ≤ (c2)!.

The specific bound is useless in practice and even the Brauer-Fowler argument is only of value in a few small cases. Nevertheless the theorem asserts that for any finite group H the determination of all finite simple groups with an involution centralizer isomorphic to H is a finite problem. In the ensuing decades Brauer, Janko and their students would show that the problem was not only finite but tractable. This suggested a two-step strategy for the proof of the Classification Theorem:

Step 1: Determine all possible structures for an involution centralizer in a finite simple group.

Step 2: For each possible structure, determine all finite simple groups with such an involution centralizer.

Brauer had proved some sample cases for Step 2. No one had a clue how to do Step 1. Indeed no one had a clue how to show that a nonabelian finite simple group even contains an involution. Or did they? Just a few months after the Congress, on December 24, 1954, a historic paper was submitted by Suzuki [Su2]


to the Bulletin of the A.M.S. It contained a beautiful proof of the nonexistence of nonabelian simple CA-groups of odd order. This was the first breakthrough in the direction of the Miller-Burnside conjecture concerning the solvability of groups of odd order. Suzuki’s CA-group proof had two parts. As mentioned above, a rather easy “local” analysis gives the structure of the maximal subgroups of a CA-group G.

Then a brilliant application of exceptional character theory leads to a contradiction.

Thompson writes [T4]:

“Suzuki’s CA-theorem is a marvel of cunning.... Once one accepts this theorem as a step in a general proof, one seems irresistibly drawn along the path which was followed [in the Odd Order Paper].”

Nevertheless the difficulties for the proof of the Miller-Burnside Odd Order Con- jecture still seem insuperable. In Suzuki’s case, the maximal subgroups of G are easily seen to be Frobenius groups. In the case of an arbitrary finite simple group of odd order, one could assume by induction that every maximal subgroup was a solvable group. But the possible structures of solvable groups are far more elabo- rate than the structures of Frobenius groups. Without major reductions using local analysis, the requisite character theory would be unthinkably difficult.

The next breakthrough came from an unexpected direction. There was consid- erable activity in the 1950’s on the Burnside and Restricted Burnside Problems, including Kostrikin’s proof of the Restricted Burnside Theorem for groups of prime exponent. In this context Philip Hall and Graham Higman wrote a remarkable pa- per [HH] in 1956 aimed at the reduction of the Restricted Burnside Problem to the prime-power exponent case. They provided an insightful analysis of the structure of finite p-solvable groups (a class containing all solvable groups). In particular they established the following results.

Hall-Higman Lemma. Let G be a finite p-solvable group and let X be the largest normal subgroup of order prime to p. Let P be a p-subgroup of G such that XP/X is the largest normal p-subgroup of G/X and let V = P/Φ(P ). Then H = G/XP acts by conjugation as a faithful p-solvable group of linear operators on the vector space V and H has no nontrivial normal p-subgroup.

Hall-Higman Theorem B. Let H be a p-solvable group of linear operators on a finite-dimensional vector space V over a field of characteristic p. Suppose that H has no nontrivial normal p-subgroup. If x∈ H with xp = 16= x, then either the minimum polynomial of x in its action on V is (t−1)p or the following conclusions hold: p is a Fermat prime, H has a nonabelian Sylow 2-subgroup and the minimum polynomial of x is (t− 1)p−1.

The Hall-Higman theorems conveyed some important messages. First of all, if G is a complicated finite solvable group (in particular, far from nilpotent), then G has a comparatively uncomplicated normal subgroup N = Op0p(G) with the property that G/N is isomorphic to a group of linear operators on a vector space V arising as a quotient of N . Secondly, Theorem B shows how linear algebra can be exploited to analyze the structure of G/N and it displays a distinction between the structure of p-solvable groups and the structure of many parabolic subgroups in groups of Lie type in characteristic p. For example, if H is the stabilizer of a 1-space in GL(n + 1, p), N = Op0p(H) and Z = Z(H) = Op0(H), then N = Z× V , where V = Op(H) which may be thought of as an n-dimensional vector space over Fp. Moreover H = H/N is isomorphic to GL(n, p) acting naturally on V . Thus


H has no nontrivial normal p-subgroup, but if p is odd, then for many x∈ H of order p, the minimum polynomial of x in its action on V has degree less than p.

Indeed the transvections have quadratic minimum polynomial (t− 1)2. Of course H is not a p-solvable group when p > 3. (When p = 3, we are precisely in one of the exceptional cases of Theorem B: H has a normal quaternion subgroup and 3 is a Fermat prime.)

Thus far the work of the 1950’s could be regarded as the brilliant belated ful- fillment of the work of the period 1890–1910. Indeed Chevalley and Steinberg had created the analogue over finite fields of the Lie theory of Killing and Cartan, com- pleting the work of Dickson. Brauer, Suzuki and Feit had extended the character theory of Frobenius and Burnside and its applications to the classification of small simple groups. The study of Zassenhaus groups by Feit, Ito and Suzuki was nearing completion, tapping out a vein of permutation group theory going back to Jordan.

Some pregnant possibilities lay in the papers of Suzuki and Hall-Higman. Never- theless there was some truth to Brauer’s assertion at the International Congress of Mathematicians in 1970 on the occasion of Thompson’s receiving the Fields Medal:

“...up to the early 1960’s, really nothing of real interest was known about general simple groups of finite order.”

4. Enter John Thompson

At the suggestion of Marshall Hall, Thompson attacked in his dissertation [T1]

the long-standing conjecture that the Frobenius kernel is always nilpotent. This is equivalent to the following assertion:

Thompson’s Thesis. Let G be a finite group admitting an automorphism α of prime order with CG(α) = 1. Then G is a nilpotent group.

If G is nilpotent, then for every prime p dividing|G|, G has a normal subgroup of index p. Thus it is natural to attack the problem via transfer. Choosing an α-invariant Sylow p-subgroup P of G, induction applies to the normalizer of char- acteristic subgroups of P . It follows from Gr¨un’s Second Theorem [Gu] that Z(P ) cannot be weakly closed in P . This led Thompson to the study of weak closures of abelian subgroups of P .

Definition. Let A≤ H ≤ G. The weak closure of A in H with respect to G is

W =hAg: Ag≤ Hi.

Equivalently, W is the smallest subgroup of H containing A and weakly closed in H (with respect to G).

Thompson’s analysis is quite delicate but eventually leads to a Hall-Higman- type situation involving a pair of elementary abelian p-groups A and B which normalize each other but do not commute. Thus [A, B] 6= 1 but [A, B] ≤ B and so [A, B, B] = 1. Hence some element x of B viewed as a linear operator on A has quadratic minimum polynomial. Thanks to freedom in the choice of p, this contradicts the Hall-Higman Theorem B. In fact the quadratic action of x permits a more elementary contradiction, but the shadow of Hall-Higman is definitely visible.

Thompson’s thesis had immediate implications for the study of Zassenhaus groups, clarifying as it did the structure of Frobenius kernels. Even more im- portant, it was the beginning of Thompson’s profound analysis of the structure


of solvable subgroups of finite simple groups. In the summer of 1958 while work- ing on his thesis, Thompson visited Wielandt in T¨ubingen, and Huppert reports Wielandt’s comment:

“Das is ein verdammt scharfsinniger Bursche. Man kann etwas lernen von ihm.”

[“That’s one damn sharp guy. You can learn something from him.”]

Thompson completed his theorem and sent his work to Philip Hall in Decem- ber 1958. Hall immediately grasped the import of Thompson’s achievement and suggested a reformulation of one of the main theorems which liberated it from the context of groups with operators.

Theorem. Let K be a finite group, p an odd prime and P a Sylow p-subgroup of K. Suppose that K 6= XP for any normal p0-subgroup X. Then there exists a char- acteristic subgroup D of P of nilpotence class at most 2 such that NK(D)/CK(D) is not a p-group.

Further consideration of weak closure and the Hall-Higman Theorem B led Thompson to the discovery of the J -subgroup and the Thompson factorization theorems. There are two slightly different definitions of the J -subgroup. I shall give the one which has become more popular recently.

Definition. Let P be a finite p-group and let d be the maximum rank of an elemen- tary abelian subgroup of P . LetA(P ) denote the set of all elementary subgroups of P of rank d. Then the Thompson subgroup J (P ) is

J (P ) =hA : A ∈ A(P )i.

Let H be a finite solvable group whose Fitting subgroup F is a p-group. For R any p-group denote by Ω1(R) the subgroup generated by the elements of order p in R. If P is a Sylow p-subgroup of H, then Z(P )≤ Z(F ) by Fitting’s Theorem and so

V = VH=hΩ1(Z(Ph)) : h∈ Hi

is a subgroup of Ω1(Z(F )). In particular V may be regarded as an H-module on which F acts trivially. The subgroup C which is the kernel of the H-action on V may be larger than F , but it shares with F the property that H/C has no nontrivial normal p-subgroup. Thus H/C may be regarded as a solvable subgroup of GL(V ) with no nontrivial normal p-subgroup, exactly the Hall-Higman setup.

Under suitable additional hypotheses, for example that H has odd order, Thompson shows that no A∈ A(P ) can act nontrivially on V , i.e. J(P ) ≤ C. But then Sylow’s Theorem immediately yields the Thompson Factorization:

H = CNH(J (P )) = CH(Ω1(Z(P )))NH(J (P )).

When hypotheses such as solvability and odd order are dropped, the analysis be- comes much more complicated, but the fundamental philosophy remains the same.

Definition. A finite group G is of (local) characteristic p-type if the following condition is satisfied by every p-local subgroup H of G: Let F be the largest normal p-subgroup of H. Then CH(F )≤ F .

Whenever G is a group of characteristic p-type, Thompson’s analysis may be undertaken. Having chosen a Sylow p-subgroup P of G, it shows that there are two p-local subgroups of fundamental importance: C = CG(Ω1(Z(P ))) and N =


NG(J (P )). If H = NG(D) for some nonidentity normal subgroup D of P and H6=

(H∩C)(H∩N), then VHmust be a “failure of factorization module” for H/CH(VH).

As stated here, this is a linguistic tautology, but in fact it has mathematical content.

In particular, failure of factorization modules are “quadratic modules”, and around 1970, Thompson [T3] classified all “quadratic pairs” for p ≥ 5, i.e. pairs (H, V ) where V is a faithful irreducible quadratic module for H in characteristic p and H is generated by its elements of order p having quadratic minimum polynomial.

Necessarily H is a product of groups of Lie type in characteristic p and V is a tensor product of small modules. This has been extended to the case p = 3, where new examples arise such as spin modules for the spin covers of the alternating groups and the Leech lattice mod 3 for the Conway group and some of its subgroups. When p = 2, all involutions act quadratically, but the analysis of failure of factorization modules remains meaningful and is at the heart of the classification of simple groups of characteristic 2-type.

Glauberman’s discovery of the ZJ -Theorem [Gl2] around 1967 provided an easier approach than factorization theorems in the context of groups of odd order. Re- cently Stellmacher [Sl1] has established an analogue of the ZJ -Theorem for groups of order prime to 3. However, in the general context, when the primes 2 and 3 are intertwined in the subgroup structure of G, Thompson’s more robust factorization approach returns to center stage.

By 1959 when Marshall Hall published his text The Theory of Groups [Ha], he could write in dramatic contrast to Brauer’s remarks in 1954: “Current research in Group Theory, as witnessed by the publications covered in Mathematical Reviews, is vigorous and extensive.” Hall invited Thompson to Caltech in the summer of 1959 and they extended Suzuki’s theorem on CA-groups of odd order to the nilpotent centralizer case. They sent a copy of the manuscript to Feit, who substantially im- proved the character theory. This launched the collaboration of Feit and Thompson on groups of odd order. Suzuki’s CA-paper was 10 pages in length. The CN-paper [FHT] was 17 pages. Feit and Thompson estimated it would take about 25 pages to prove the Odd Order Theorem. That proved to be quite an underestimate.

Fundamental to the analysis of the minimal simple group G of odd order as pursued in the Odd Order Paper is the dichotomy between those groups G in which the intersection M∩ Mg is always “small”, for M a maximal subgroup of G and g∈ G − M, and those groups G in which the intersection is sometimes “large”.

The former case was primarily to be handled by extending the Brauer-Suzuki- Feit analysis of exceptional characters to an even more complicated setting. The local methods Feit and Thompson introduced to treat the latter case had profound impact on the rest of the Classification. By contrast the Odd Order Paper was the high watermark for character theory in the Classification. Future applications were few and far less intricate. Because of the greater resonance of the local methods in future papers, I shall discuss them along with related generalizations in the remainder of this section.

In their attack on the “large” case, Feit and Thompson were motivated by an important paper of Philip Hall [H5] from 1956, extending his earlier work on criteria for solvability in terms of permutability of Sylow subgroups. Hall introduced the symbol Epqto denote the existence in G of a Hall{p, q}-subgroup. As Thompson’s notes in [T4], Hall’s 1956 paper “suggests that a group is solvable if and only if it satisfies Ep,q for all primes p, q.” Thompson undertook to prove such E-theorems for groups of odd order. He writes:


“From my work on the Frobenius kernel, I wanted to work upwards from the bottom, and the bottom is the Fitting subgroup. So, in trying to prove Ep,q, I could see the value in trying to locate q-subgroups Q which are normalized by a Sylow p-subgroup of G.”

This led to the fundamental concept of A-signalizers.

Definition. Let A and H be subgroups of the group G with A a p-group and let π be a set of primes not containing p. The set IH(A; π) is the set of all A-invariant π-subgroups of H, and IH(A; π) is the set of maximal elements of IH(A; π) under inclusion.

Thus in this language Thompson was seeking Q ∈ IG(P ; q) for P a Sylow p- subgroup of G. If H is an A-invariant solvable p0-group, then by Philip Hall’s theorem, the members of IH(A; π) are Hall π-subgroups of H and are transitively permuted by CH(A). Thompson saw how to extend this result to the context of local solvability.

Thompson Transitivity Theorem. Let G be a finite group in which every p- local subgroup is solvable. Let A be an abelian p-group of rank at least 3 which is a Sylow p-subgroup of CG(A). Then CG(A) transitively permutes the elements of I

G(A; q) for all primes q6= p.

If P is a Sylow p-subgroup of G containing A, then as NG(A) permutes the elements of IG(A; q) with CG(A) acting transitively, Lagrange’s Theorem yields that NP(A) normalizes some Q ∈ IG(A; q). Bootstrapping upward often yields some Q∈ IG(P ; q), as Thompson desired.

The requirement that A have p-rank at least 3 is unavoidable and makes precise the above-mentioned subdivision of the analysis into small and large cases, not only in the Odd Order Paper but throughout the Classification. The Transitivity Theorem is a key tool in the proof of the Uniqueness Theorem below, which is the basic result for eliminating the “large” case:

The Feit-Thompson Uniqueness Theorem. Let G be a finite group of odd or- der in which every proper subgroup is solvable. Suppose that K is a proper subgroup of G such that either r(K)≥ 3 or r(CG(K))≥ 3. Then K is contained in a unique maximal subgroup of G. (Here r(K) denotes the maximum rank of an abelian p- subgroup of K, as p ranges over all prime divisors of|K|.)

A beautiful proof of the Uniqueness Theorem, incorporating Glauberman’s ZJ - Theorem and his own new ideas, was discovered by Bender [Be1] in the late 1960’s.

Around the same time, Gorenstein and Walter undertook a profound analysis of the Feit-Thompson signalizer arguments with the goal of extending the analysis to a context where p-local subgroups are no longer solvable. They solved this problem with the concept of an A-signalizer functor, which evolved over the years, the following elegant definition being due to Goldschmidt [Go2].

Definition. Let A be an abelian p-subgroup of the finite group G. Then a function θ mapping the set A#into the set of solvable A-invariant p0-subgroups of G is called a solvable A-signalizer functor if θ(a) = θ(CG(a))≤ CG(a) for all a∈ A# and the following “balance equation” holds for all a,b∈ A#:

θ(CG(a))∩ CG(b) = θ(CG(b))∩ CG(a).


They then focussed attention on the sets Iθ(A, π) consisting of those members X of IG(A, π) such that CX(a)≤ θ(CG(a)) for all a∈ A#. Finally Iθ(A) is the union of all of these sets, and the Solvable Signalizer Functor Theorem states:

Solvable Signalizer Functor Theorem. Let G be a finite group, A an abelian p-subgroup of G with r(A)≥ 3 and θ a solvable A-signalizer functor. Then Iθ(A) has a unique maximal element θ(G).

Early versions of this theorem were proved around 1969 by Gorenstein [G2]. The full theorem for p = 2 was established by Goldschmidt [Go3], and then finally the general case was proved by Glauberman [Gl4] in 1973. The proof entails remaining in the realm of solvable θ-subgroups of G where it is possible to recover some of Thompson’s results, such as the Transitivity Theorem. A Nonsolvable Signalizer Functor Theorem was proved in the late 1970’s by McBride [McB1], [McB2].

Once θ(G) exists, the fundamental dichotomy is:

θ(G)6= 1 or θ(G) = 1.

When θ(G) 6= 1, the simplicity of G implies that M = NG(θ(G)) is a proper subgroup of G containing the normalizers of many p-subgroups of G. We call such a subgroup a p-uniqueness subgroup. In many contexts, such as the Odd Order Pa- per, θ(G) = 1 implies that G is of characteristic p-type and Thompson factorization analysis may be pursued. In the Odd Order Paper the absence of failure of fac- torization modules again leads to the existence of p-uniqueness subgroups. This is an important milestone, but considerable difficult character theory and generator- and-relations arguments are still necessary to complete the proof of the Odd Order Theorem.

Adrian Albert organized a Group Theory Year at the University of Chicago in 1960-61. This was perhaps the most successful mathematical year ever organized.

Feit and Thompson completed most of their work on the Odd Order Theorem.

Suzuki pursued his research on 2-transitive permutation groups. Gorenstein and Walter began their collaborative study of groups with dihedral Sylow 2-subgroups.

The first real signs appeared of the evolving sociology of the classification effort as a team project. (The earlier Brauer-Suzuki-Wall paper was an instance of par- allel research arriving at approximately the same point at the same time.) What also became evident was the unprecedented scale of the undertaking. The Odd Order Paper was an unbelievable 255 pages in length, and there was no fat on the manuscript. Many of the later manuscripts would follow this triple digit pattern, culminating in the 731 page Memoirs volume by Gorenstein and Lyons [GL1] and the forthcoming 800+ page Quasithin Paper by Aschbacher and S. D. Smith [AS].

Feit and Thompson published the Odd Order Paper in 1963 [FT]:

The Odd Order Theorem. All finite groups of odd order are solvable.

This short sentence and its long proof were a moment in the evolution of finite group theory analogous to the emergence of fish onto dry land. Nothing like it had happened before; nothing quite like it has happened since. I compare the character theory (Chapters 3 and 5) to Bach’s B Minor Mass, the glorious summation of everything which had been achieved by Frobenius, Brauer, Suzuki and Feit himself, on the theme of wresting information about group structure from the arithmetic of induced characters. It is hard to imagine pushing this analysis through in a more complicated setting. Luckily no one ever needed to. By contrast, I compare the


local analysis (Chapter 4) to Beethoven’s Ninth Symphony. Looking back at earlier group theory it declared: “O Freunde, nicht diese T¨one!” Looking ahead to the great collaborative effort of the next 40 years whose size and shape it presaged, it declared: “Sondern lasst uns angenehmere anstimmen!”

5. Back to the prime 2

Once the Odd Order Paper was completed, attention naturally focussed on the prime 2. The work of Brauer, Suzuki and Wall in the 1950’s had shown how the existence of involutions could be exploited to characterize simple groups starting from fairly detailed 2-local data. The Odd Order Paper (and Thompson’s evolving work [T2] on minimal simple groups of even order) were the only models for arriving at such detailed local data. The dichotomy which had emerged in the Odd Order Paper between groups of p-rank at most 2 and those of p-rank at least 3 suggested the importance of groups of 2-rank 2 as a separate problem. (A 2-group of rank 1 is either cyclic or quaternion. An old argument (sketched earlier) shows that no simple group of even order except C2 has cyclic Sylow 2-subgroups. Brauer and Suzuki [BS] had proved that no simple group has quaternion Sylow 2-subgroups.) Gorenstein and Walter [GW1] completed the dihedral case. Luckily, an elegant argument of Alperin showed that a 2-group of 2-rank 2 which was a candidate to be a Sylow 2-subgroup of a simple group must fall into one of four infinite families (dihedral, semidihedral, wreathed, homocyclic abelian) or be of one exceptional isomorphism type. Alperin’s proof was made possible by an elegant application of modular character theory to 2-fusion analysis by Glauberman [Gl1]:

Glauberman’s Z-Theorem. Let G be a finite group with no nontrivial normal subgroup of odd order. Let z be an involution of G. Either z∈ Z(G) or z commutes with a G-conjugate zg with zg6= z.

Alperin and Gorenstein undertook the semidihedral/wreathed case, which turned out to be quite difficult and to require considerable 2-modular character theory, for which they turned to Brauer. By 1969 simple groups of 2-rank at most 2 had been classified [ABG1], [ABG2]. Once it could be assumed that G had 2-rank at least 3, it was at least possible to begin 2-signalizer functor analysis. This was not, however, quite enough to pass gracefully from the conclusion of the Signalizer Functor Theorem for p = 2 to the existence of a 2-Uniqueness Subgroup of the desired type when θ(G)6= 1. To bridge this gap Gorenstein and Harada produced a monumental work [GH] classifying simple groups of sectional 2-rank at most 4.

Later Harada [Hr1] discovered a short and elegant argument to build the same bridge, though their magnum opus was quoted in many other contexts.

When a 2-uniqueness subgroup M exists, there remains the problem of identi- fying the group G. The obvious examples are the groups SL(2, 2n), P SU (3, 2n) and Sz(2n), which arise as the conclusions of Suzuki’s Theorem on 2-transitive permutation groups. The strongest form of 2-uniqueness subgroup can be defined in permutation group language as:

Definition. A group G has a strongly embedded subgroup M if G is a transitive permutation group of even order in which every involution fixes exactly one point.

(M is the stabilizer of a point.)


One obvious gap between this definition and Suzuki’s hypotheses is that G is not assumed to be 2-transitive. Indeed this need not be the case if G has 2-rank 1.

However, Bender [Be3] succeeded in 1968 in proving the following beautiful theorem:

Bender’s Strongly Embedded Theorem. Let G be a finite simple group with a strongly embedded subgroup M . Then G satisfies the hypotheses of Suzuki’s The- orem. Thus G ∼= SL(2, 2n), Sz(2n) or P SU (3, 2n).

In 1973 building on work of Gorenstein and Walter and of Shult, Aschbacher [A1] was able to strengthen Bender’s Theorem to the precise 2-Uniqueness Theorem needed for the Signalizer Functor Method.

Paralleling these developments was another line of research which yielded dra- matic surprises. Beginning early in the 1960’s several researchers began to analyze simple groups with abelian Sylow 2-subgroups. It soon became clear that a crucial case was when the centralizer of an involution z has the form:

CG(z) =hzi × L ∼= Z2× P SL(2, q)

for some q ≡ ±3 (mod 8). Thompson thought he could show that necessarily q = 32n+1, a case arising in the simple Ree groups 2G2(32n+1), but Janko [Ja]

discovered that one further case was possible: q = 5. Janko’s discovery of J1 (a subgroup of G2(11)), the first new sporadic simple group in a century, rivetted the attention of the group theory world and began a decade of feverish exploration and discovery in which 20 more sporadic groups came to light. The strategy of studying promising involution centralizers soon led Janko to two more simple groups with an isomorphic involution centralizer, and shortly thereafter rewarded Held, Lyons and O’Nan with simple groups. The construction of Janko’s second group, J2, by M. Hall as a rank 3 permutation group inspired a flurry of constructions of new simple groups by D. Higman and Sims, McLaughlin, Suzuki and Rudvalis (the latter a prediction, followed by a construction by Conway and Wales). Conway’s investigations of the Leech lattice and Fischer’s study of 3-transposition groups (discussed below) each led to three new sporadic simple groups.

A crowd of new Ph.D.’s entered the fray in the late 1960’s and the classification project entered high gear. The level of excitement is captured in the language of Gorenstein’s introduction to his book Finite Groups [G1], published in 1968:

“In the past ten years there has been a tremendous surge of activity in finite group theory. The period has witnessed the first serious classification theorems concerning simple groups and the discovery of several new families of simple groups;

and, above all, the fundamental question of the solvability of groups of odd order has been answered. ... Out of the work of Feit and Thompson ... and Suzuki..., there is gradually emerging a body of techniques and a series of general methods for studying simple groups. Although the entire field is presently in an excited state of ferment and fluidity, as recent basic work of Glauberman and Alperin clearly indicates, a degree of stability appears to be settling over certain aspects of the subject.”

In 1969, John Walter [Wa] achieved the reduction of the problem of groups with abelian Sylow 2-subgroups to the specific centralizer of involution problem which had been studied by Thompson, Janko and others. In this work he was forced to analyze the 2-signalizer problem in a context of nonsolvable involution centralizers. Soon he and Gorenstein began their deep analysis of the Signalizer


Method, leading not only to the concept of a signalizer functor discussed above but to other fundamental concepts of balance and generation.

The search for a suitable replacement for the Fitting subgroup, which could be regarded as the “bottom” of a nonsolvable group led Gorenstein and Walter [GW2]

to the p-layer and extended p-layer. Inspired by comments of Gorenstein, Bender [Be2] was led to the generalized Fitting subgroup.

Definitions. 1. A finite group L is quasisimple if L = [L, L] and L/Z(L) is a simple group. A finite group E is semisimple if E is the commuting product of quasisimple groups.

2. If H is a finite group, then E(H) is the unique maximal normal semisimple subgroup of H. The generalized Fitting subgroup of H is F(H) = E(H)F (H).

3. If H is a finite group, p is a prime and Op0(H) is the largest normal p0- subgroup of H, then the p-layer of H, Lp0(H), is the subgroup of the full preimage of E(H/Op0(H)) generated by p-elements.

The generalized Fitting subgroup F(G) plays the role of the foundation on which G stands, in the following sense:

Bender’s F-Theorem. Let H be any finite group. Then F(H) is the commut- ing product of the semisimple group E(H) and the nilpotent group F (H). Moreover F(H) contains CH(F(H)). Thus H/Z(F (H)) acts faithfully as a group of auto- morphisms of F(H) and, in particular |H| ≤ |F(H)|!.

For the structure theory of finite groups, F(G) is the key concept. On the other hand for the purpose of studying the embedding of a 2-local subgroup H in a finite group G, the 2-layer of H is more important, because it enjoys the following fundamental “balance” property, discovered by Gorenstein and Walter [GW3].

L-Balance Theorem (Gorenstein-Walter). Let H be a 2-local subgroup of the finite group G. Then L20(H)≤ L20(G).

This crucial result together with various extensions and modifications facilitates the comparison of the centralizers of two different commuting involutions in a finite group G. Its proof relies fundamentally on a weak version of Schreier’s Conjecture, which was proved by Glauberman as a corollary of his Z-Theorem. A posteriori we know the truth of the full Schreier Conjecture and hence the validity of the analogous Lp0-Balance Theorem for all primes p.

In addition to these major theorems, the Signalizer Functor Method requires a good choice of signalizer functor. A candidate functor was proposed by Gorenstein and Walter. Later Goldschmidt introduced a better functor, which was further modified and implemented by Aschbacher in his characterizations of simple groups of Lie type. By 1971 the stage was mostly set for the final attack on CFSG. Or was it?

6. Gorenstein’s Classification Program

On the one hand major conceptual advances in the understanding of the lo- cal structure of finite simple groups had been achieved. On the other hand the great classification theorems of the 1960’s from the Odd Order Theorem through the Alperin-Brauer-Gorenstein Theorem, while consuming almost 1,500 journal pages, had only completed the characterization of the groups P SL(2, q), P SL(3, q), P SU (3, q), Sz(2n) and 2G2(3n) (the split BN -pairs of rank 1 in the language of



好了既然 Z[x] 中的 ideal 不一定是 principle ideal 那麼我們就不能學 Proposition 7.2.11 的方法得到 Z[x] 中的 irreducible element 就是 prime element 了..

Wang, Solving pseudomonotone variational inequalities and pseudocon- vex optimization problems using the projection neural network, IEEE Transactions on Neural Networks 17

volume suppressed mass: (TeV) 2 /M P ∼ 10 −4 eV → mm range can be experimentally tested for any number of extra dimensions - Light U(1) gauge bosons: no derivative couplings. =>

For pedagogical purposes, let us start consideration from a simple one-dimensional (1D) system, where electrons are confined to a chain parallel to the x axis. As it is well known

The observed small neutrino masses strongly suggest the presence of super heavy Majorana neutrinos N. Out-of-thermal equilibrium processes may be easily realized around the

incapable to extract any quantities from QCD, nor to tackle the most interesting physics, namely, the spontaneously chiral symmetry breaking and the color confinement.. 

(1) Determine a hypersurface on which matching condition is given.. (2) Determine a

• Formation of massive primordial stars as origin of objects in the early universe. • Supernova explosions might be visible to the most