Table Of ContentThe Status of the
Classification of the
Finite Simple Groups
Michael Aschbacher
Common wisdom has it that the theorem Assume G is finite, and write H (cid:2) G to indi-
classifying the finite simple groups was cate that H is a normal subgroup of G. A normal
proved around 1980. However, the proof seriesfor G is a sequence
of the Classification is not an ordinary 1=G (cid:1) G (cid:1) ··· (cid:1) G =G,
0 1 n
proof because of its length and com-
plexity, and even in the eighties it was a bit con- and the family of factors of the series is the
troversial. Soon after the theorem was established, family
Gorenstein, Lyons, and Solomon (GLS) launched a (G /G :0≤i<n)
i+1 i
program to simplify large parts of the proof and,
perhaps of more importance, to write it down of factor groups arising in the sequence. Observe
clearly and carefully in one place, appealing only the series is maximal (i.e. we cannot adjoin an extra
to a few elementary texts on finite and algebraic term between Giand Gi+1) precisely when each fac-
tor is simple. The maximal series are called com-
groups and supplying proofs of any “well-known”
position series, and we have:
results used in the original proof, since such proofs
were scattered throughout the literature or, worse, Jordan-Hölder Theorem. All composition series
did not even appear in the literature. However, the for G have the same length and the same (un-
GLS program is not yet complete, and over the last ordered) family of simple factors.
twenty years gaps have been discovered in the
The simple factors in a composition series for
original proof of the Classification. Most of these
G are called the composition factorsof the group.
gaps were quickly eliminated, but one presented
They do not determine G up to isomorphism, but
serious difficulties. The serious gap has recently
they do exert a lot of control over the gross struc-
been closed, so it is perhaps a good time to review
ture of G. Thus the simple groups are analogous
the status of the Classification. I will begin slowly
to the primes in number theory, although one does
with an introduction to the problem and with some
not quite have “unique factorization”.
motivation.
Recall that a group G is simple if 1 and G are Example. Recall G is solvable if each of its com-
the only normal subgroups of G; equivalently position factors is of prime order. Recall also that
G∼= G/1 and 1∼= G/G are the only factor groups Galois’s Theorem says this is the class of groups
of G. corresponding to polynomials solvable by radicals.
Further, by a result of Philip Hall, solvable groups
Michael Aschbacher is professor of mathematics at the Cal- satisfy an important generalization of Sylow’s The-
ifornia Institute of Technology. His email address is orem, and indeed this property characterizes solv-
asch@its.caltech.edu. able groups. This generalization says: If G is solv-
This work was partially supported by NSF-0203417. able of order n and m is a divisor of n with
736 NOTICESOFTHEAMS VOLUME51, NUMBER7
(m,n/m)=1, then G has a subgroup of order m, make the following observation: With the exception
all subgroups of G of order mare conjugate in G, of some of the sporadic groups, each group G ap-
and each subgroup of order dividing m is con- pearing in the theorem can be regarded as essen-
tained in some subgroup of order m. tially the group of automorphisms of some fairly
accessible mathematical object X. The existence of
One can conceive of an analysis of the finite
X gives an existence proof for G, but, more im-
groups based on a solution to the following two
portant, the representation of Gon Xgives a means
problems:
for studying G and obtaining information about
The Classification Problem. Determine all finite representations of the simple groups in various cat-
simple groups. egories, most particularly representations as per-
mutation groups (i.e. subgroup structure of G) and
The Extension Problem.Given groups Xand Y,de-
linear groups. The verification of some special
termine all extensions of X by Y; i.e. determine all
property of such representations is usually the
groups G with a normal subgroup H such that
∼ ∼ sort of extra input necessary to apply the Classi-
H = X and G/H = Y.
fication to solve a given problem.
In practice the Extension Problem is too hard, Indeed some such information was necessary to
except in special cases. It seems better not to look prove the Classification in the first place. That is
too closely at the general finite group, but instead to say, in proving the Classification Theorem, one
when faced with a problem about finite groups, to begins with the list Kof simple groups appearing
attempt to reduce the problem or a related prob- in the statement of the theorem and considers a
lem to a question about simple groups or groups minimal counterexample to the Classification
closely related to simple groups. Then using the Theorem: A finite simple group Gminimal subject
Classification of the finite simple groups and knowl- to G∈/ K. Thus each proper subgroup J of G is a
edge of the simple groups, solve the reduced prob- K-group: if K(cid:2) H ≤Jwith H/Ksimple, then the
lem. Note this procedure works only if one knows section H/K is in K. The proof of the Classifica-
enough about simple groups to solve the problem tion depends very heavily on facts about the
for simple groups; this is where the Classification subgroup structure and linear representations of
comes in: it supplies an explicit list of groups which K-groups.
can be studied in detail using the effective de- During this article I will be discussing two dif-
scription of the groups supplied by the Classifica- ferent efforts: The “original proof” of the Classifi-
tion. cation and work done since about 1980 aimed at
This approach to solving group theoretic prob- improving, simplifying, and writing down carefully
lems has been in use since about 1980, when the in one place a proof of the Classification. The orig-
finite simple groups were deemed to have been clas- inal proof sought only to make sure the literature
sified. It has been extremely successful: virtually actually contains all the pieces of some program
none of the major problems in finite group theory purporting to prove the Classification Theorem. The
that were open before 1980 remain open today. second effort aims to produce a more readable
Moreover, finite group theory has been used to treatment that inspires a higher level of confi-
solve problems in many branches of mathematics. dence.
In short, the Classification is the most important There is a two-volume exposition of part of the
result in finite group theory, and it has become in- original proof by Gorenstein in [G1] and [G2]. Goren-
creasingly important in other areas of mathemat- stein’s books do not attempt to give details of the
ics. proof, but only to give an outline of what is entailed.
Now it is time to state the: Further, Gorenstein died before completing the
third volume of the series.
Classification Theorem.Each finite simple group
The largest part of the second effort is the pro-
is isomorphic to one of the following groups:
gram begun by Gorenstein, Lyons, and Solomon
1. A group of prime order. (GLS). While this program does not attempt to ad-
dress all parts of the proof, if completed as envi-
2. An alternating group.
sioned, it would deal with most parts. The work is
3. A group of Lie type. being published by the American Mathematical So-
ciety (AMS), and at the time I write this article, five
4. One of 26 sporadic groups.
volumes in the series have appeared in [GLS]. In
Observe that the statement of the Classifica- [GLS] you will also find references to other parts
tion Theorem given above is deceptively simple. In of the second effort.
order for it to have real content, one must define I have described the Classification as a theorem,
what one means by “group of Lie type” and “spo- and at this time I believe that to be true. Twenty
radic group”. Constraints on space preclude in- years ago I would also have described the Classi-
cluding such definitions here, so instead I will fication as a theorem. On the other hand, ten years
AUGUST2004 NOTICESOFTHEAMS 737
ago, while I often referred to the Classification as r =p from when r (cid:4)=p. To implement Step 1 we
a theorem, I knew formally that that was not the must translate these notions for linear groups into
case, since experts had by then become aware that related notions defined in abstract groups.
a significant part of the proof had not been com- Let p be a prime, G a finite group, and H a
pletely worked out and written down. More pre- p-local subgroup of G. Define H to be of charac-
cisely, the so-called “quasithin groups” were not teristic pif
dealt with adequately in the original proof. Steve C (O (H))≤O (H),
H p p
Smith and I worked for seven years, eventually
classifying the quasithin groups and closing this where Op(H) is the largest normal p-subgroup of
gap in the proof of the Classification Theorem. We H and for U ⊆G, CH(U) is the subgroup of all el-
completed the write-up of our theorem last year; ements of G commuting with each element of U.
it will be published (probably in 2004) by the AMS. Define Gto be of characteristic p-typeif each p-local
Later I will state the result; it should be viewed as subgroup of G is of characteristic p, and define G
part of both the original proof and the second ef- to be of even characteristicif for each 2-local sub-
fort. group Hof G containing a Sylow 2-subgroup of G,
It is time for some specifics: The proof of the His of characteristic 2. It turns out that each group
Classification proceeds by studying the so-called of Lie type and characteristic pis of characteristic
local subgroups of G. Let p be a prime. A p-local p-type.
subgroup of G is the normalizer of a nontrivial For reasons I will not go into, the prime 2 plays
p-subgroup of G. a special role in the local theory of finite simple
Let Gbe our minimal counterexample. The proof groups. (For example, by the Feit-Thompson The-
of the Classification can be thought of as made up orem [FT], nonabelian finite simple groups are of
of two steps: even order.) Thus in the original proof the follow-
ing partition appears:
Step 1. Prove the local structure of G resembles
that of some G¯ ∈K. Case I.The minimal counterexample G is of char-
acteristic 2-type.
Step 2. Use the resemblance in Step 1 to prove
G∼= G¯. Case II.G is notof characteristic 2-type.
Step 2 for the sporadic groups is one of the The generic group appearing in Case I is a group
parts of the second effort that [GLS] does not ad- of Lie type and characteristic 2, while almost all
dress. However, there has been quite a bit of other simple groups appear in Case II.
progress in improving the treatment of Step 2 since There is also a partition according to size that
the original proof. Many of the original proofs of corresponds roughly to the notion of size for
the existence and uniqueness of the sporadic groups of Lie type: Given a prime p and a finite
groups were machine aided, and the mathematics group G, define the p-rank mp(G) of G to be the
involved was often unnecessarily complicated. In maximal dimension of an abelian subgroup of G
the last twenty years, methods with the flavor of of exponent p, regarded as a vector space over the
combinatorial group theory and/or algebraic topol- field of order p. In Case II the “size” of G can be
ogy have emerged that provide simpler, more con- taken to be the 2-rank m2(G)of G. In Case I the size
ceptual, treatments of uniqueness in Step 2 and is the parameter e(G)defined by Thompson in the
have eliminated almost all computer calculations. n-group paper [T] (the model for all later work on
Case I):
I believe, however, that some of the old computer-
aided existence and uniqueness proofs have not e(G)=max{m (H):H isa2-localofGand
p
been superseded; e.g. the proof of the uniqueness
pisanoddprime}.
of Thompson’s group and the existence proof for
O’Nan’s group are probably still machine aided. I Define G to be quasithin if e(G)≤2. The “small”
will not say more than that about Step 2 but instead groups in Case I are the quasithin groups. Thus in
will focus on Step 1. the original proof we have four blocks in our par-
The generic finite simple group is a finite group tition corresponding to the large and small groups
of Lie type. Each such group G is described via a of characteristic 2-type and to the large and small
representation as a linear group, say G≤GL(V)for groups not of characteristic 2-type. The quasithin
some finite-dimensional vector space V over some groups of characteristic 2-type constitute one of the
finite field F. Thus G has a characteristic which is four blocks.
a prime: The characteristic pof F. Similarly the Lie In the GLS program, one of the partitions is
rank of G is a measure of the “size” of G, and for changed by altering the definition of “characteris-
our purposes we can think of the Lie rank as roughly tic”. I will not give the GLS definition, since it is tech-
the dimension of V. Finally, given a prime r, the r- nical. However, to accommodate the GLS change,
local structure of G is qualitatively different when Steve Smith and I also work with a different notion
738 NOTICESOFTHEAMS VOLUME51, NUMBER7
of characteristic in our study of quasithin groups: class of groups in the extended Case I have at least
Recall that G is of even characteristic if been overcome for the small groups in Case I.
CH(O2(H))≤O2(H) for each 2-local containing a In the original proof of the Classification, chrono-
Sylow 2-subgroup of G. Define a finite group G to logically Case II was treated before Case I, more time
be a QTKE-groupif G is quasithin of even charac- was spent dealing with Case II, and more people
teristic and each proper simple section of G is in worked on this case. Probably as a result, the treat-
K. ment of Case II in the original proof is in better
shape than the treatment of Case I. Volume 6 in the
Classification of QTKE Groups.(Aschbacher-Smith)
[GLS] series treats the small groups in (their rede-
Let G be a nonabelian simple QTKE-group. Then G
fined) Case II. After a certain point, GLS treat the
is isomorphic to one of the following:
large groups in Cases I and II together; this work
1. A group of Lie type of characteristic 2 and Lie rank is begun in volume 5 of [GLS] and will be completed
at most 2, but not U5(q). in the next few volumes. Thus the part of the sec-
ond effort that is furthest from completion is the
2. L4(2), L5(2), Sp6(2), or U5(4). initial phase of Case I.
The approach used to treat large groups in
3. The alternating group A9.
Case I is to focus on p-locals for odd primes pbut
4. L2(p) for p a Mersenne or Fermat prime, L3(3), to keep 2-locals in the picture. This leads us to the
U3(3), L4(3), U4(3), or G2(3). following definitions: Given a 2-local H, define
5. One of 11 sporadic groups: a Mathieu group, a σ(H)={p:pisanoddprimeandm (H)>2},
p
Janko group other than J1, HS, He, or Ru.
and let σbe the union of the sets σ(H), as Hvaries
The proof of this theorem will be published by
over the 2-locals of G. We work with p-locals for
the AMS in two volumes in [AS]. It is roughly 1,200 p∈σ.
pages in length. In part this reflects the complex-
In the original proof, two problems in Case I re-
ity of the proof, but it also reflects a style of ex-
quired special treatment:
position that includes more detail than one usually
A. The uniqueness case: There exists a 2-local H
finds in the original proof of the Classification, with σ(H)(cid:4)=∅and Hstrongly p-embedded in Gfor
and it reflects our decision to keep our treatment each p∈σ(H); that is, |H∩Hg| is prime to pfor
as self-contained as possible. Indeed, one of the two each g∈G−H.
volumes is devoted to group-theoretic infrastruc- B. The case e(G)=3.
ture, such as proofs of folk theorems and facts Initially GLS hoped that new methods devel-
about K-groups. oped after 1980 (e.g. the so-called amalgam
To my knowledge the main theorem of [AS] method) could be used to treat both cases (A) and
closes the last gap in the original proof, so (for the (B) for the larger class of groups of even type, and
moment) the Classification Theorem can be re- their program was based on the assumption that
garded as a theorem. On the other hand, I hope I specialists in those methods would handle the two
have convinced you that it is important to complete cases. However, to date this has not happened, so
the program by carefully writing out a more reli- the treatment of cases (A) and (B) in groups of
able proof in order to minimize the chance of other even type remains as perhaps the greatest obsta-
gaps being discovered in the future. Thus our dis- cle to the completion of the second effort. The
cussion of the status of the Classification would not work on these cases in groups of characteristic
be complete without some indication of what re- 2-type in the original proof is available as a model,
mains to be done in that program. but it is not clear how much more difficult the
Recall that the condition that Gbe of even char- problems are in groups of even type. Gernot Stroth
acteristic is weaker than the condition that G be and Inna Korchagina have done preliminary work
of characteristic 2-type; thus if one changes the par- on (A) and (B), respectively, under the hypothesis
tition in the original proof to a division based on that Gis of even type. Case (B) requires special treat-
the even characteristic condition, more groups ap- ment, because different signalizer functors (cf.
pear in Case I. GLS work with so-called groups of Volumes 1 and 2 of [GLS]) are used in p-rank 3 and
even type; again, this condition is weaker than the p-rank greater than 3.
characteristic 2-type condition, so more groups Finally, in addition to the original proof of the
also appear in Case I in their partition, and hence Classification and the second-generation approach
this part of the problem becomes more difficult. of GLS, there is also a third-generation program in-
However, as a corollary to our main theorem, Steve volving a number of people, particularly Ulrich
Smith and I also determine in [AS] the quasithin Meierfrankefeld, Bernd Stellmacher, and Gernot
groups of even type, the result on quasithin groups Stroth, that would treat all groups of characteris-
required in the GLS approach to the Classification. tic 2-type (and perhaps eventually all groups of even
Thus any difficulties involved in treating the larger characteristic) using the amalgam method. Steve
AUGUST2004 NOTICESOFTHEAMS 739
Smith and I made use of variations of this method
AMS SHORT COURSE
in our work in [AS]. It is possible that this approach
will give a better treatment of Case I or at least of
Case (B).
The Radon Transform and
Applications to Inverse Problems References
[AS]M. ASCHBACHERand S. SMITH, The Classification of Qua-
sithin Groups, Math. Surveys Monogr., Amer. Math.
Atlanta, Georgia, January 3-4, 2005
Soc., Providence, RI, to appear.
[FT]J. THOMPSONand W. FEIT, Solvability of groups of odd
order, Pacific J. Math.13(1963), 775–1029.
[G1]D. GORENSTEIN, Finite Simple Groups; An Introduction
to Their Classification, Plenum, New York, 1982.
[G2] ———, The Classification of the Finite Simple Groups,
Volume I, Plenum, New York, 1983.
[GLS]D. GORENSTEIN, R. LYONS, and R. SOLOMON, The Classi-
fication of the Finite Simple Groups, vol. 40, Math. Sur-
veys Monogr., Amer. Math. Soc., Providence, RI; Num-
ber 1: 1995, Number 2: 1996, Number 3: 1997, Number
4: 1999, Number 5: 2002.
[T]J. THOMPSON, Nonsolvable finite groups all of whose local
subgroups are solvable, Bull. Amer. Math. Soc. (N.S.)
74 (1968), 383–437; II, Pacific J. Math. 33 (1970),
451–536; III, Pacific J. Math.39(1971), 483–534; IV, Pa-
Organizers: cific J. Math.48(1973), 511–592; V, Pacific J. Math.50
•Gestur Olafsson, Louisiana State University (1974), 215–297; VI, Pacific J. Math.51(1974), 573–630.
•Todd Quinto, Tufts University
Speakers:
•Liliana Borcea, Rice University
•Adel Faridani, Oregon State University
•Peter Kuchment, Texas A&M University
•Alfred Louis, Universitaet des Saarlandes
•Peter Massopust, Tuboscope Pipeline Services
•Todd Quinto, Tufts University
Tomagraphy is important in pure and applied mathematics,
as well as in several branches of applied sciences, in partic-
ular diagnostic radiology, nondestructive evaluation, and
other forms of image reconstruction. The Short Course
will cover the basic mathematics behind tomography and
will describe important applications. The talks will be aimed
at a general audience, beginning with elementary facts about
the Radon transform and then introducing important current
research areas, including impedance imaging, local tomog-
raphy, wavelet methods, regularization and approximate in-
verse, and emission tomography. Several special sessions at
the AMS meeting will continue the themes introduced in the
Short Course.
740 NOTICESOFTHEAMS VOLUME51, NUMBER7
