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Home , Finite group, Ronald Solomon, Simple group

The Classification of Finite Simple Groups: A Progress Report Ronald Solomon

Communicated by Alexander Diaz-Lopez and Harriet Pollatsek

Danny Gorenstein dubbed the classification of finite simple groups “The Thirty Years’ War.”

Ronald Solomon is professor emeritus at the . His email address is [email protected]. For permission to reprint this article, please contact: [email protected]. DOI: http://dx.doi.org/10.1090/noti1689

646 Notices of the AMS Volume 65, Number 6 History of the Project is one significant change worthy of note. (A detailed In 1981 the monumental project to classify all of the fi- overview of the original Classification Project may be nite simple groups appeared to be nearing its conclusion. found in [ALSS].) Danny Gorenstein had dubbed the project the “Thirty We anticipate that there will be twelve volumes in the Years’ War,” dating its inception from an address by complete series [GLS], which we hope to complete by at the International Congress of Mathe- 2023. I will report on the current state of our project in maticians in 1954. He and Richard Lyons agreed that it this article. would be desirable to write a series of volumes that would contain the complete proof of this Classification Theorem, Introduction to Classification modulo a short and clearly specified list of background re- A common theme in is to study a partic- sults. As the existing proof was scattered over hundreds ular mathematical structure and attempt to classify all of journal articles, some of which cited other articles instances of it, e.g., regular polyhedra, distance transitive that were never published, there was a consensus that graphs, and 3-, among others. Classifying infi- this was indeed a worthwhile project, and the American nite groups is hopeless, but already in the 1890s, inspired Mathematical Society (AMS) agreed to publish this series by the work of Killing and Cartan on finite-dimensional of volumes. In the spring of 1982, Danny and Richard semisimple complex Lie , some mathematicians recruited me to be a partner in this project. Richard Foote began to contemplate the classification of all finite groups. and Gernot Stroth were also recruited at an early stage to Can we classify all finite groups? Since we can combine contribute specific portions of this work. any two finite groups, for instance via direct product, to create a new finite , it is natural to begin by considering groups that are “building blocks” for all other finite groups. Similar to the factorization of into prime numbers, one can “break down” finite groups into smaller pieces called simple groups, which cannot be decomposed further. A group 퐺 is simple if it has no nontrivial proper normal ; i.e., its only normal are the trivial group and 퐺 itself. The Jordan-Hölder Theorem then tells us that for any finite group 퐺, there is an ordered sequence of subgroups, 1 = 퐻1 ◁ 퐻2 ◁ ⋯ ◁ 퐻푛 = 퐺 called a , such that each 퐻푖 is a maximal proper of 퐻푖+1 and all 퐻푖+1/퐻푖, called composition factors, are simple. Moreover, any two composition series of a group 퐺 are equivalent in the sense that they have the same number of subgroups and the same composition factors, and . Nonetheless, the problem of determining all ways to reassemble a set of composition factors into a finite group is daunting, perhaps infeasible. Although it is very easy to prove that the only abelian simple groups are ℤ/푝ℤ, where 푝 is a , the problem of determining all finite 푝-groups, i.e., groups 퐺 all of whose composition factors Richard Lyons (pictured here with the author) and are isomorphic to ℤ/푝ℤ, is of frightening complexity. For 10 Gorenstein agreed that it would be desirable to write example, there are billions of groups of size 2 , and a series of volumes [GLS] that would contain the many of them are essentially indistinguishable. complete proof of this Classification Theorem. Fortunately, in most applications of finite , where the group arises as a group of or Considerable progress was made during the first or linear operators on some other structure, decade of the project, and then, tragically, Danny Goren- the problem can be reduced easily to the case where stein died in August 1992. Nevertheless, the first six the is “primitive” in some sense. Using the volumes of our series were published by the AMS during classification of finite simple groups, Bob Guralnick, the the decade 1994–2005. Then a hiatus ensued. I am happy latest recipient (see Notices, April 2018, p. 461), to report that Volume 7 has just been published by the has demonstrated the efficacy of this strategy in a wide AMS, and Volume 8 is near completion and promised to variety of contexts. the AMS by August 2018. The completion of Volume 8 For the remainder of this article, I will discuss the will be a significant mathematical milestone in our work. classification of the finite simple groups. In contrast to It seems a good time to provide this progress report on the abelian case, the classification of nonabelian finite the GLS project. Moreover, although the strategy outlined simple groups is quite complex and requires a full- in Volume 1 [GLS] remains substantially unchanged, there scale classification strategy. The two smallest nonabelian

June/July 2018 Notices of the AMS 647 simple groups are 퐴5, the on five by a theorem of Borel and Tits they inherit significant symbols, with 60 elements, and PSL(2, 7), a member of structural properties from these parabolics. On the other one of the families of groups of Lie type (see the next hand, this is far from true in groups of Lie type when 퐹 has section), with 168 elements. In addition to the simple odd , with a small number of interesting exceptions. groups belonging to infinite families, there are also This gave rise to the initial definitions of “even” and “odd”: twenty-six so-called sporadic groups, five discovered by characteristic 2-type and “non”-characteristic 2-type. (We Mathieu in the nineteenth century, and the rest discovered shall instead call the latter “odd type.”) between 1965 and 1975. The sporadic groups range in size Definition. A group 퐻 is of 2-parabolic type if 퐻 contains from 7920 (the smallest ) to approximately a normal 2-subgroup 푄 such that the centralizer 퐶 (푄) is 8 × 1053 (the aptly named Monster). 퐻 just 푍(푄). (This definition is nonstandard. It is used here In the next section, we discuss our classification for expository purposes.) strategy to prove the following theorem. Definition. A group 퐺 is of characteristic 2-type if every Theorem. Every finite is isomorphic to one 2-local subgroup of 퐺 is of 2-parabolic type. of the following: (1) a of prime order, A weakness of using this definition to define the line (2) an alternating group of degree at least 5, between even and odd is that, in the study of groups (3) a simple , or of odd type, it forces the consideration of all groups 퐺 (4) one of the 26 “sporadic” simple groups. having a 2-local, no matter how small, which is not of 2-parabolic type. A better definition is The Classification Strategy Definition. A group 퐺 is of even characteristic (or para- For about fifty years, the Classification Strategy has been bolic characteristic 2) if every 2-local subgroup 퐻 of odd schematically represented as a box subdivided into four index in 퐺 is of 2-parabolic type. smaller boxes: In the GLS Project, we use another term, “even type,” small odd small even which is convenient for our approach but technically large odd large even complicated to define. Recently, Magaard and Stroth have classified all groups that are of even type but not of Most of the finite simple groups are groups of Lie type even characteristic. The list is rather short: 퐴 , Ω (3), defined over some finite field. If you are not familiar with 12 7 Ω−(3), and the sporadic simple groups 퐽 , 퐶표 , and 퐹푖 . Lie groups, you may consider the example of PSL(푛, 퐹), 8 1 3 23 I shall speak henceforth of the odd/even dichotomy as which is the quotient of the group SL(푛, 퐹) of 푛 × 푛 the dichotomy between groups of even type and groups matrices of determinant 1 with entries from the finite of odd (i.e., not even) type. field 퐹 by the normal subgroup of scalar matrices of The small/large dichotomy was first formulated for determinant 1. In this context, odd and even refer to groups of even type by John G. Thompson in his work the of |퐹|, while small and large are measured on 푁-groups, i.e., nonsolvable finite groups all of whose by the size of 푛. When 푛 < 3, the groups are definitely local subgroups are solvable. He defined the parameter small, though PSp(4, 퐹) is also small. If you know Tits’s 푒(퐺) to be the maximum rank of an abelian subgroup of description of groups of Lie type as BN pairs, then it is odd order contained in some 2-local subgroup of 퐺.A more accurate to say that groups of BN rank 1 or 2 are finite simple group 퐺 is of small even type if 푒(퐺) ≤ 2, small, while most groups of BN rank at least 3 are large. i.e., if no 2-local subgroup 퐻 of 퐺 contains a subgroup Of course, there are also simple alternating groups and isomorphic to 퐶 × 퐶 × 퐶 for any odd prime 푝. Such sporadic groups which must be fitted into this scheme. 푝 푝 푝 groups also came to be known as quasithin groups, Much more serious is the fact that we must provide and the classification of quasithin simple groups of definitions for terms like “a group 퐺 of small odd type” even characteristic by Aschbacher and Smith [AS] was the that do not presuppose that 퐺 is a group of Lie type culminating accomplishment of the original Classification or indeed has any known property other than simplicity. Project. Notice that if 퐺 is a Chevalley group defined over However, if 푝 is a prime divisor of |퐺|, then Sylow a field of even order 푞 > 2, then 푒(퐺) typically measures guarantees the existence of many 푝-subgroups of 퐺, i.e., the rank of a split torus of 퐺 and hence the BN-rank of 퐺. subgroups of order 푝푚 for some 푚 ≥ 1. We call the The small/large dichotomy for groups of odd type was centralizers and normalizers of such subgroups 푝-local originally formulated in terms of the 2-rank (or normal subgroups of 퐺. In his 1954 address, Richard Brauer 2-rank or sectional 2-rank) of 퐺. In the [GLS] volumes, the made the case for attempting to characterize simple study of groups 퐺 of odd type focuses on the isomorphism groups via their 2-local structures. A crucial validation types of components of involution centralizers, where an for this strategy was provided in 1963 when involution is an element of order 2 in 퐺. The following and John G. Thompson [FT] published their proof that all definitions are of crucial importance. nonabelian finite simple groups have even order. Now, if 퐺 = 퐺(퐹) is a group of Lie type defined Definitions. over a field F of even order, then the 2-local subgroups 1. A finite group 퐿 is quasisimple if 퐿/푍(퐿) is a non- of 퐺 are contained in parabolic subgroups of 퐺, and abelian simple group and 퐿 = [퐿, 퐿].

648 Notices of the AMS Volume 65, Number 6 and no 2-local subgroup 퐻 of 퐺 contained a subgroup isomorphic to 퐶푝×퐶푝×퐶푝, unless 푝 = 3. It was anticipated that the Quasithin Problem would be enlarged to treat this wider problem. Given the monumental task accomplished by Aschbacher and Smith, it is quite understandable that they limited themselves to the original Quasithin Problem (though extended to groups of even type). It should also be noted that both Aschbacher–Smith [AS] and [GLS] treat only 풦-proper simple groups, i.e., simple groups all of whose proper simple sections are known simple groups. This certainly suffices for an inductive proof of the Classification Theorem. In the philosophy of the GLS Project, large simple groups of both even and odd type are to be treated as groups of type, i.e., groups 퐺 that, for a Some of those who have worked on the Classification suitable prime 푝, have 푝-rank at least 3 and have elements Project (l to r: David Goldschmidt, Ron Solomon, of order 푝 whose centralizers have components of suitable Stephen D. Smith, Richard Lyons, , type. Notice that if 퐺 is a simple group of Lie type defined John G. Thompson, and Richard Foote), who gathered over a field 퐹 of characteristic 푟, then centralizers of at Cambridge University in 2013 in celebration of elements of order 푟 have no components, while elements Thompson’s eightieth birthday. of prime order 푝 different from 푟 often do. For example, if 퐺 = GL(6, 퐹), |퐹| = 22푚, and 푥 is an element of 퐺 of order 3 with three 2-dimensional eigenspaces, with eigenvalues 2 2. A subgroup 퐿 of a group 퐻 is a component of 퐻 if 퐿 1, 휔, and 휔 , then 퐶GL(6,퐹)(푥) has three components, is a quasisimple subnormal subgroup of 퐻. (Subnormality each isomorphic to SL(2, 퐹) as in Figure 2. Again, the is the transitive extension of the normality relation.) calculations are similar in SL(6, 퐹) and in PSL(6, 퐹). If, for example, 퐺 = GL(푉), where 푉 is an 푛-dimensional 퐼 푂 푂 vector space over a finite field 퐹 of odd order, and if 푡 푥 = ⎜⎛푂 휔퐼 푂 ⎟⎞ is an involution in 퐺, then the components of 퐶 (푡) 2 퐺 ⎝푂 푂 휔 퐼⎠ typically are SL(푉−(푡)) and SL(푉+(푡)), where 푉−(푡) and and 푉+(푡) are the −1 and +1 eigenspaces for 푡 on 푉. Thus the components of 퐶GL(푉)(푡) are [퐶+, 퐶+] ≅ SL(푉+) and [퐶−, 퐶−] ≅ SL(푉−). See Figure 1. (Computations are ⎪⎧ 퐴 푂 푂 ⎪⎫ similar in SL(푉) and PSL(푉), and the components are 퐶 (푥) = ⎜⎛푂 퐵 푂⎟⎞ ∶ 퐴, 퐵, 퐶 ∈ GL(2, 퐹) GL(6,퐹) ⎪⎨ ⎪⎬ isomorphic to those in GL(푉).) ⎩⎝푂 푂 퐶⎠ ⎭ ≅ GL(2, 퐹) × GL(2, 퐹) × GL(2, 퐹). 퐼 푂 푡 = ( ) 푂 −퐼 Figure 2. The centralizer 퐶GL(6,퐹)(푥) of an order-3 element 푥 has three components, each isomorphic to 퐴 푂 2푚 퐶 (푡) = {( ) ∶ 퐴 ∈ GL(푉 ), 퐵 ∈ GL(푉 )} SL(2, 퐹). Here |퐹| = 2 . GL(푉) 푂 퐵 + − In general, in groups 퐺 of characteristic 푟, centralizers = 퐶+ × 퐶− ≅ GL(푉+) × GL(푉−). of elements of order 푝 will typically have components that themselves are quasisimple groups of Lie type of Figure 1. The components of 퐶GL(푉)(푡) are characteristic 푟, indeed often smaller dimensional ver- [퐶+, 퐶+] ≅ SL(푉+) and [퐶−, 퐶−] ≅ SL(푉−). sions of 퐺 itself, as in the example in Figure 2. Thus, the Roughly speaking, groups of small odd type are defined principal criterion for “components of suitable type” is to be simple groups 퐺 such that the only components that they not be groups of Lie type of characteristic 푝. In occurring in any involution centralizer are isomorphic to this case, we shall call 푝 a semisimple prime for 퐺. SL(2, 퐹) or PSL(2, 퐹) for some field 퐹 of odd order. The More accurately, we choose 푝 = 2 if some involution classification of finite simple groups of small odd type centralizer 퐶퐺(푡) has a component which is not a group of appears in Volume 6 [GLS]. Since the [GLS] Project will Lie type in characteristic 2. We also exclude most sporadic quote [AS] for the classification of quasithin groups of components and, for the sake of “largeness,” SL2(푞) and even type, 2004 marked the publication of the volumes L2(푞). If we cannot choose 푝 = 2, then we choose an treating “small” groups of both odd and even type. odd prime 푝 such that 퐺 has 푝-rank at least 3 and some It should be noted here that in the outline of the [GLS] 푝-element centralizer 퐶퐺(푥) has a component which has a series in Volume 1, a slightly broader definition of small noncyclic Sylow 푝-subgroup and is not a group of Lie type even type was formulated. Namely, it was stipulated that 퐺 in characteristic 푝. (For 푝 = 3, we also exclude several was of small even type if 퐺 was of even type with 푒(퐺) ≤ 3 sporadic components.)

June/July 2018 Notices of the AMS 649 I have swept under the carpet a serious problem concerning “푝-signalizers,” and I shall continue to do so. But I will say that I should be saying “푝-components” and not “components” in the previous paragraph, and the proof that we may reduce the problem from 푝-components to components when either 푝 = 2 or 푝 is odd and 퐺 has 푝-rank at least 4 is one of the principal results of Volume 5 [GLS]. For this reason, we use the term generic even type to refer to the subset of groups of large even type which have 푝-rank at least 4 for a suitable odd prime 푝. The signalizer problem for groups of characteristic 2-type and odd 푝-rank 3 was handled by Aschbacher in a pair of papers, which still need to be generalized to the case of groups of even type. Building on the work in Volume 5 [GLS], Volumes 7 and 8 will complete the proof of the following two theorems. Theorem O. Let 퐺 be a finite 풦-proper simple group of odd type. Then either 퐺 is an alternating group of degree 푛 ≥ 5 (but not 8 or 12) or 퐺 is a group of Lie type defined over a finite field of odd order or 퐺 is one of the following ′ sporadic simple groups: 푀11, 푀12, 퐽1, 푀푐, 푂 푁, or 퐿푦. Theorem GE. Let 퐺 be a finite 풦-proper simple group of generic even type. Then either 퐺 is a group of Lie type defined over a finite field of even order or 퐺 has a proper 푝-uniqueness subgroup for some odd prime 푝 or 퐺 has a “푝-thin configuration” for some odd prime 푝. Volume 7 [GLS] has just been published by the AMS. It almost completes the identification of the alternating groups of degree 푛 ≥ 13 and the reduction of the Generic Case to the case where a suitable 푝-element centralizer has a component of Lie type in characteristic 푟, with either Gernot Stroth’s 600-page treatment of groups with a strongly 푝-embedded subgroup will probably appear as Volume 11 [GLS].

푝 = 2 or 푟 = 2. Volume 8 will complete the identification of the alternating groups and the generic groups of Lie type. It is near completion, and we anticipate submission to the AMS for publication by August 2018. An important project stimulated by our work was the study of a class of amalgams of finite groups, known as Curtis–Tits–Phan amalgams, by a team of mathematicians including C. Bennett, R. Blok, R. Gramlich, C. Hoffman, M. Horn, B. Muhlherr, W. Nickel, and S. Shpectorov. Their results are added to our list of Background Results and used in our identification of many of the finite simple groups of Lie type. An extensive overview of this project, with a fuller list of authors and papers, may be found in Gramlich [Gr]. A very recent paper of Blok, Hoffman, and Shpectorov is also crucial to our work. What will remain to be done after Volume 8 [GLS]? The remaining work falls primarily into three cases: (1) the Bicharacteristic Case, (2) the 푝-Uniqueness Case, and (3) the 푒(퐺) = 3 Case. Inna Capdeboscq, coauthor of Volume 9 [GLS]. The Bicharacteristic Case refers to the case when 퐺 is of even type with 푒(퐺) ≥ 4 but there is no semisimple prime

650 Notices of the AMS Volume 65, Number 6 푝. Thus, 퐺 seems to have two different characteristics: It would be wonderful to complete our series by 2023, 2 and 푝. Using an argument of Klinger and Mason, it is the sixtieth anniversary of the publication of the Odd easy to see that 푝 = 3. This is a phenomenon that occurs Order Theorem. Given the state of Volumes 8, 9, 10, and in many sporadic simple groups as well as a few groups 11, the achievement of this goal depends most heavily on defined over fields of order 2 or 3. Considerable work has the completion of the 푒(퐺) = 3 problem. It is a worthy been done on this problem, originally by Gorenstein and goal. Lyons, and more recently by Inna Capdeboscq, Lyons, and me. We anticipate that this will be the principal content Credits of Volume 9 [GLS], coauthored with Capdeboscq. This Photo of Danny Gorenstein and group photo of Classification volume will also dispose of the dangling 푝-Thin case from Project participants courtesy of Richard Foote. Theorem GE. Photo of Richard Lyons and Danny Gorenstein by Richard Foote. The 푝-Uniqueness Case arises when the proof of The- Figures 1 and 2 by author. orem GE leads to the construction of a proper subgroup Photo of Inna Capdeboscq by Yves Capdeboscq. 푀 of 퐺 containing all (or almost all) of the 푝-local over- Photo of Gernot Stroth courtesy of Gernot Stroth. groups of a fixed Sylow 푝-subgroup of 퐺. When 푝 = 2, Author photo by Rose Solomon. this problem was handled elegantly by Helmut Bender and Aschbacher, and their work is contained in Volume References 4 [GLS]. (Similarly, the 2-Thin Problem was handled in [ALSS] M. Aschbacher, R. Lyons, S. D. Smith, and R. Solomon, Volume 6, which explains why the statement of Theorem The Classification of Finite Simple Groups, Mathematical O is so much cleaner than the statement of Theorem GE.) Surveys and Monographs Vol. 172, Amer. Math. Soc., 2011. When 푝 is odd, there is a major 600-page manuscript by MR2778190 [AS] M. Aschbacher and S. D. Smith, The Classification of Gernot Stroth treating groups with a strongly 푝-embedded Quasithin Groups I, II, Mathematical Surveys and Mono- subgroup, which will appear in the [GLS] series, probably graphs 111, 112, Amer. Math. Soc., 2004. MR2097623, in Volume 11. There are also substantial drafts by Richard MR2097624 Foote, Gorenstein, and Lyons for a companion volume [FT] W. Feit and J. G. Thompson, Solvability of groups of odd (Volume 10?), which together with Stroth’s volume will order, Pacific J. Math. 13 (1963), 775–1029. MR0166261 complete the 푝-Uniqueness Case. In particular, Volumes [GLS] D. Gorenstein, R. Lyons, and R. Solomon, The Classifi- 9, 10, and 11 will strengthen Theorem GE to cation of the Finite Simple Groups, Mathematical Surveys and Monographs 40, 1–7, Amer. Math. Soc., 1994, 1996, Theorem GE+. Let 퐺 be a finite 풦-proper simple group of 1998, 1999, 2002, 2005, 2018. MR1303592, MR1358135, generic even type. Then 퐺 is a group of Lie type defined MR1490581, MR1675976, MR1923000, MR2104668 over a finite field of even order. [Gr] R. Gramlich, Developments in finite Phan theory, Innov. Incidence Geom. 9 (2009), 123–175. MR2658896 There remains the gap between generic even type and large even type, i.e., the 푒(퐺) = 3 Case. If 퐺 is a 풦-proper simple group of characteristic 2-type with 푒(퐺) = 3, then 퐺 was classified in a pair of papers by ABOUT THE AUTHOR Michael Aschbacher. The first paper treats the case when Ronald Solomon first learned to a 2-local 퐻 contains an abelian 푝-subgroup 퐴 of rank love words from his mother; to 3 for some 푝 > 3. Aschbacher’s method in this case love math from his geometry should extend fairly easily to our case, i.e., the case when teacher, Blossom Backal; and to 퐺 is of even type. In the second paper, 3 is the only love classical music from Mozart’s choice for 푝. This is the case that was redefined as a Great G minor Symphony. He is the subproblem of the Quasithin Problem in Volume 1 [GLS]. proud husband of Rose, father of It has been given serious attention by several of our Ari and Michael, and grandfather Ronald Solomon colleagues, notably Capdeboscq and Chris Parker, who of Sofia. have a manuscript overcoming one of the difficulties of extending Aschbacher’s work to the even type situation. In this context, it might be very helpful to have a proof of the following result: Statement. Let 퐺 be a finite group with an abelian Sylow 푝-subgroup 푃. Suppose that 푃 contains a strongly closed elementary abelian subgroup 퐵. Then 퐺 contains a normal subgroup 푁 such that 퐵 = ⟨푥 ∈ 푃 ∩ 푁 ∶ 푥푝 = 1⟩. Using the Classification Theorem, this result is a special case of a theorem of Flores and Foote. However, we are not permitted to quote the Classification Theorem. For the purposes of our problem, it would suffice to assume that 푝 = 3 and 푃 has 3-rank 3.

June/July 2018 Notices of the AMS 651