The Enormous Theorem Author(s): Daniel Gorenstein Source: Scientific American, Vol. 253, No. 6 (December 1985), pp. 104-115 Published by: Scientific American, a division of Nature America, Inc. Stable URL: http://www.jstor.org/stable/24967876 Accessed: 16-10-2017 21:46 UTC

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This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC All use subject to http://about.jstor.org/terms The Enormous Theorem'

The classification of the finite, simple groups is unprecedented in the history of mathematics, for its proof is 15, 000 pages long. The exotic solution has stimulated interest far beyond the field

by Daniel Gorenstein

ow could a single mathematical forms a group; indeed, the rules for is called closure. Furthermore, to be theorem require 15,000 pages combining the members of a group a group the operation * must satisfy to prove? Who could read are simply borrowed, in more general three rules. First, the set must include a suchH a proof? Who could pass judg­ form, from some of the rules of ordi­ so-called identity element, designated ment on its validity? Yet there it is: the nary arithmetic. Given the striking ap­ e, such that for any element a in the set proof that all finite, simple groups have plicability of arithmetic in daily life, it the products a * e and e * a are equal to been found has run to between 10,000 is hardly surprising that the same con­ a. Second, for each element a in the set and 15,000 pages. Of course, no one cepts in a more abstract setting have there must be some element in the set person is responsible for the achieve­ become powerful tools for the under­ called the inverse of a and designated ment, nor is the size of the proof attrib­ standing of the universe. a-I, such that the products a * a-I and utable to lengthy computer calcula­ The fundamental building blocks a-I * a are equal to e. Finally, the oper­ tions (although computers are used at for all groups are the simple groups. ation * must be associative. In other one place in the analysis). The work is Such groups bind together like the at­ words, for every three members a, b instead the combined effortof more oms in a molecule to generate ever and c in the set the sequence in which than 100 mathematicians, primarily more intricate patterns of groups. The the operation * is carried out does not from the U.S., England and Germany, term "simple" has no structural con­ affect the result: a * (b * c) is equal to but also from Australia, Canada and notation: just as the atom has a highly (a*b)*c. Japan. The complete proof is scattered complex internal structure, so does the Note that the definition does riot re­ across the pages of some 500 articles simple group. A simple group is simple quire the operation * to be commuta­ in technical journals, almost all pub­ only in the sense that, like the atom, it tive: a * b need not in general be equal lished between the late 1940's and the cannot be decomposed into smaller en­ to b * a. In fact, most of the interesting early 1980's. tities of the same kind. groups arising in mathematics as well The idea of a mathematical group Many groups have an infinite num­ as in nature are noncommutative [see has been fundamental in mathemat­ ber of elements, or members; again the illustration on page 109]. ics since it was first introduced by the number system is a good example, for One can now appreciate how the French mathematician Evariste Ga­ there is no limit to the size of the num­ rules for combining the elements in a lois in the 1830's. Galois used prop­ bers that can be added together. The group are the basic laws of arithmetic erties of groups to settle in the negative rules for combining group elements in more abstract form. Thus consider a question that was then 200 years old: are also satisfied, however, by many how the integers form a group when Can the solutions of polynomial equa­ systems of finite size; such systems are the combining operation * is taken to tions of the fifth degree and higher be called fihite groups. For example, the be ordinary addition. The group prod­ expressed by formulas similar to the hours registered by a clock can be add­ uct of two integers a and b is then their familiar one for quadratic equations ed like numbers, but the result is ex­ ordinary sum; for example, 5 * 7 is (and the less familiar ones for third­ pressed as an hour between one and 12: 5 + 7 or 12. Moreover, the identity el­ degree and fourth-degree equations)? five hours later than 10 o'clock is three ement is the number 0, the inverse of Although the application Galois had o'clock, not 15 o'clock. The 12 hours the number a is the number -a and for in mind was limited to polynomial together ,with clock addition form the any numbers a, band c the associative equations, the concept of a group has so-called clock group with 12 ele­ law holds; for example, 3 + (4 + 5) is turned out to occur throughout both ments. Clock groups having "periods" equal to (3 + 4) + 5. Similarly, the mathematics and nature. other than 12 can also be readily con­ nonzero rational numbers together The rotations of a sphere, the peri­ structed [see illustration on page 106]. with ordinary multiplication also form odicities of a crystal and the sym­ a group: the number 1 is the identity metries of atoms are all examples of hat then is a group? The formal element and the number 1/ a is the in­ groups. Even the interactions of ele­ Wdefinition is precise, but it is per­ verse of the number a. (The number 0 mentary particles and the "eightfold haps best understood through specific must be excluded because it has no in­ way," whereby certain particles are de­ examples. A group is a set of elements verse.) The 12 hours of a clock form a scribed as composite objects made up together with an operation designated group in which zero o'clock, that is, 12 of quarks, need group-theoretical for­ by an asterisk ( *) for combining them. o'clock, is the identity element and the mulations in order to be understood. Thus for any two elements a and b in inverse of, say, five o'clock is seven The system of ordinary integers to­ the set, the "product" a * b must also o'clock. These examples of arithmetic gether with the operation of addition be a member of the set. This property groups are commutative because nei- This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 104 All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC ther the addition nor the multiplica­ There is an analogy between group number is not prime, it is composite, tion of numbers depends on the order theory and ordinary mUltiplication and the so-called fundamental theo­ in which the numbers are combined. that can help in picturing how com­ rem of arithmetic states that every On the other hand, groups whose ele­ plex groups are built up from simple composite number can be factored ments are certain rotations or reflec­ groups. Every whole number is either into a unique set of prime numbers. tions of regular geometric figures such prime or composite. A prime number The composite number 12, for exam­ as spheres or equilateral triangles are such as 2,3,5,7 and 11 is evenly divisi­ ple, has the prime factors 2, 2 and 3. almost always noncommutative. ble only by itself and 1. If a whole In group theory there is a process

a b c d

e g h

j k

m n o p

SYMMETRY OPERATIONS on the regular dodecahedron form a fixed direction in space depends on the order in which the rota­ a mathematical object called a group, which has been a major topic tions are carried out. There are 60 elements in the group, which of interest to mathematicians for the past 150 years. The theory of correspond to the five positions of each of the 12 faces of the do­ groups has also found important applications outside mathematics, decahedron; the illustration shows each face in the front position notably in crystallography and in the physics of elementary parti­ (a-I) and the five front positions of the dark blue face (I-p). The cles. All finite groups are built up from so-called simple groups, group has played a major role in the history of mathematics. Eva­ which playa role in group theory similar to the role of atoms in riste Galois, the mathematician who invented group theory, showed physics or prime numbers in arithmetic. The rotations of the do­ that the simplicity of this group and the fact that it has a nonprime decahedron that preserve its orientation in space form the smallest number of elements lead to the resolution of a classic mathematical simple group whose elements do not commute with one another: in problem: the general polynomial equation with rational coefficients other words, the finalposition of the dodecahedron with respect to whose highest-power term is x5 cannot be solved with radicals.

This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC lOS All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC called telescoping that splits an arbi­ set of simple groups can be combined includes exactly 8 X 9 X 10 X 11, or trary finite group into a unique set of to form many nonequivalent groups. 7,920, elements. The sixth sporadic simple groups in much the same way group, which includes exactly 17 5,560 as a composite number is decomposed t is not only the extraordinary length elements, was uncovered by Zvoni­ into its prime factors. Moreover, the I of the proof that has made the clas­ mir Janko, then at Monash Univer­ number of elements in each of these sification of the finite, simple groups so sity in Australia, a full century after simple-group components is a factor unusual in the annals of mathematics Mathieu's work. of the number of elements in the par­ but also the intriguing nature of the Thereafter the strange new sporadic ent group, and the product of all these solution. As research on the problem creatures began to pop up at the rate of numbers is equal to the number of ele­ progressed, group theorists not only about one per year, keeping pace with ments in the parent group. were discovering infinite families of the intense theoretical developments One must be careful, however, not to simple groups-and in the end a total of the 1960's and 1970's. The excite­ push the analogy too far. Indeed, the of 18 such families were constructed­ ment of the discoveries spilled over to simple-group components of a parent but also were discovering a number of the larger mathematical community. group may contain a composite num­ highly irregular groups that fitted into The climax came in 1982, when Rob­ ber of elements. Even more significant, none of the regular families. The first ert Griess, Jr., then at the Institute for whereas the product of all the prime five of these puzzling sporadic simple Advanced Study, constructed a group numbers in a given set (such as the set groups, as they came to be called, had that came to be known as the monster, 2, 2, 3, 5) is a unique composite num­ been found by Emile Mathieu in the because the number of elements in it ber (in this case 60), in general a given 1860's; the smallest Mathieu group is 808,017,424,794,512,875,886,459,- 904, 961,710, 757, 005,754, 368, 000,- 3 000,000, or roughly 8 X 105 . Ulti­ GROUP IDENTITY INVERSE mately 26 sporadic simple groups were SET OPERATION ELEMENT ELEMENT discovered. The monster is the largest ORDIN ARY ADDITION 0 -a of them, but because its structure ex­ INTEGERS [2 3 5; [3 + 0 OJ [3 (-3) = OJ hibits so many internal symmetries, O. ±1. ±2 .... 3 + -4J + + (-7)= = = Griess renamed it the friendly giant. NONZERO MULTIPLIC ATION 1 Finding a simple group is one thing, -.L RATIONAL plq and discovery is its own reward, but NUM BERSplq x = [ x 1 - [� � fsJ � -gJ3 proving that you have them all is quite [2 X 1 -1] 3 v,- another matter. That is the assertion of the classification theorem: The uni­ 12 HOURS ADDITION oO'CLOCK 12-aO'CLOCK ON A CLOCK MODULO 12 verse of finite, simple groups is made [3 O'CLOCK [20'CLOCK 12 + up of the 18 regular, infinite families of 11 1 = + = O'CLOCK (12-2)O'CLOCK or o groups and the 26 sporadic groups, 10 0 2 + 30'CLOCKJ OO'CLOCKJ [G Q=(} and no others! It is the proof of this 9 3 347 8 4 statement that has taken the 500 arti­ 5 cles and nearly 15,000 journal pages. 7 6 + Q 0=8J In the normal course of events an 682 explicitly conjectured result may long precede its proof. The conjecture is not 11 HOURS ADDITION OO'CLOCK 11 -a O'CLOCK ON A CLOCK MODULO 11 made without excellent reason: the re­ [30'CLOCK [20'CLOCK + 11 + sult is suggested by prior investiga­ 10 1 OO'CLOCK = (11-2 )O'CLOCK = or tions, or there exist mathematically 9 2 + 0 30'CLOCKJ OO'CLOCKJ [8 Q=(} nontrivial examples for which the con­ 8 3 3 4 7 jecture is valid. Throughout most of 4 7 6 5 + the war game against the finite, simple 0 0=8 groups, however, group theorists were 6 8 3 J unable even to estimate the size of the 3 HOURS ADDITION OO'CLOCK 3-aO'CLOCK enemy troops. It was plausible to think ON A CLOCK MODULO 3 that many-perhaps a great many­ [2O'CLOCK [20'CLOCK + + sporadic simple groups remained bun­ oO'CLOCK = (3 -2)O'CLOCK = [0+Q=C) 20'CLOCKJ OO'CLOCKJ kered out of sight. Consequently it was 2 1 0 not until many years after work on the 2 1 classification problem had begun that + = group theorists were able even to for­ 0 V v QJ mulate the theorem they hoped even­ tually to prove. EXAMPLES OF GROUPS are given in the table. Each group is made up of a set of ele­ ments and a group operation, customarily called group multiplication, whereby the ele­ his reality forced a cautious strate­ ments of the set can be combined to yield another (usually a third) group element. The group Tgy. Battles were fought over only operation, which can be designated by au asterisk (*), acts much like ordinary multiplication limited types of simple groups. These or addition. It must obey the associative law: for any elements a, band c in the group the hard-won victories yielded restricted triple group "product" (a * b) * c must be equal to the triple product a * (b* c). The set of ele­ classification theorems, which gradu­ ments must include an identity element, designated e, such that e and e the group g * * g, ally began to root out the smaller of products of any element with the identity element, are equal to For each element in the g g. g the remaining sporadic groups. The set there must also be an inverse element g-l such that g * g-l and g-l * g, the group prod­ e. process is well illustrated by Janko's ucts of g with its inverse, are equal to the identity element The set of elements in a group can be infinite, as it is in the first two exampIes, or finite, as it is in the last three. The fourth discovery of the sixth sporadic group, and fifth examples are the so-called clock groups of period 11 and period 3 respectively; which is related to the 17 th regular their elements are added as if they were the hours on an II-hour or a three-hour clock. family of simple groups that had been

106 This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC found by Rimhak Ree of the Universi­ ty of British Columbia in 1960. 1 2 6 8 0 Within every simple group there are 5 I I certain smaller groups, called central­ - x I izers of involutions, that are important for understanding the structure of the 3 4 7 8 '+ 10 6 parent group. For the Ree groups the centralizers of involutions can be rep­ resented by two-by-two square ma­ "MULTIPLICATION" of two two-by-two square matrices is carried out according to the trices, whose four entries are drawn procedure shown. Similar rules exist for multiplying two n-by-n square matrices for any from a finite number system of size number n. The matrix product is a square matrix that has the same number of entries as the equal to some odd-numbered power two multiplying matrices. The entries of the matrices can be drawn from the elements of of 3. For example, if the odd power any given number system, which is a mathematical system whose elements can be added, subtracted, multiplied and divided in much the same way as ordinary numbers. The ration­ of 3 is 1, the finite number system is al numbers and the complex numbers are familiar examples of infinite number systems, made up of the three elements in and the elements of clock groups of prime period generate examples of some of the possible the clock group of period 3. To prove finite number systems. In the matrices shown here the entries are drawn from the clock one of the early, restricted classifi­ group of period 11. To calculate in this finite number system one must first compute the cation theorems it was necessary to ordinary arithmetic result and then find the remainder after dividing that result by 11. For show that the Ree groups are the only example, the element 1 X 5 + 2 X 7 at the upper left in the product matrix (third matrix from simple groups that have the follow­ left) is equal to 19, and its remainder after division by 11 is 8. Matrices are often considered ing property: their centralizers of in­ as individual mathematical objects. All finite groups, and in particular all the finite, simple volutions can be represented by two­ groups, can be represented as groups of matrices that are combined by matrix multiplication. by-two matrices whose entries are drawn from a finite number system of a size equal to an odd power of some the contradiction he had initially simple group, and then explicit cal­ prime number p. It was a natural first sought. At first glance it might seem culations, analogous to experimental step toward that goal to try to prove remarkable that all possible matrix verification, proved that such a group the following conjecture: If an arbi­ products of A's and B's would be really exists. trary simple group has the above-men­ equivalent to one of only 175,560 ma­ This two-stage process is even more tioned property, the size of the finite trices, for that is precisely what it dramatically illustrated by Janko's sec­ number system from which the entries means for a group of the required kind ond success. If J1 was discovered by of the two-by-two matrices are drawn to exist. For example, matrix products "fudging" the properties of the cen­ is an odd power of the prime number 3. of more than a million of the A's and tralizers of involutions in the Ree In time this conjecture was verified,ex­ the B's must belong to the set. Indeed, groups, why not try a similar tack with cept in the single case for which p is there are roughly 1149, or 1051, seven­ some other known simple groups? Jan­ equal to 5 and the odd power is 1. by-seven matrices with entries from ko hit the jackpot almost immediately; Janko set out to study the exception­ the clock group of period 11, and so in fact, he made a double hit. From a al case of the Ree group problem with the products formed by the two gener­ single centralizer of an involution he every expectation that there were no ating matrices yield only a small frac­ found evidence for two possible new simple groups of the given type having tion of the total number of possible simple groups, one with 604,800 ele­ a number system of size 5 1, or 5. In matrices. The calculations were none­ ments and the other with 50,232,960. spite of all his efforts, however, he was theless carried out entirely by hand, This time, however, Janko was not not able to eliminate this possibility and they verified the existence of the able to provide the experimental verifi­ and thus complete the proof of the sixth sporadic group, which is now cation. Using the information derived conjecture. On the contrary, with con­ called J1 in honor of Janko. by Janko, Marshall Hall, Jr., and Da­ siderable effort he managed to dem­ vid Wales of the California Institute of onstrate that if such a simple group anko's construction of J1 exhibits a Technology constructed the smaller existed, it would have to contain ex­ j certain parallel with the physics of group, J2, and Graham Higman of actly 23 X 3 X 5 X 7 X 11 X 19, or elementary particles. Theoretical the University of Oxford and John 175,560, elements. analysis provided evidence for a new McKay of Concordia University in It was too much for Janko to believe he could establish such a sharp conclu­ sion unless an actual group were hov­ 0 0 0 0 0 0 8 2 10 10 8 10 8 ering in the background. He pushed on 0 0 0 0 0 0 9 3 3 3 with heightened expectation, arguing next that if such a group existed, it 0 0 0 0 0 0 10 10 8 10 8 8 2 would have to be generated by two 0 10 8 10 8 8 2 10 seven-by-seven matrices whose row A= 0 0 0 0 0 B=

and column entries are drawn from the 0 0 0 0 0 0 8 10 8 8 2 10 10 clock group of period 11 [see bottom illustration at right] . In other words, 0 0 0 0 0 0 3 3 9 3 if the two generating matrices are 0 0 0 0 0 0 3 3 9 3 called A and B, the group is made up of all possible matrix products, of the TWO SEVEN-BY-SEVEN MATRICES generate all the matrices that make up the ele­ two matrices, such as AA, BB, ABA, ments of the sporadic simple group discovered in 1965 by Zvonimir .Jankoof Monash Uni­ BBAABABBB and so forth. versity in Australia. The entries of the matrices are drawn from the elements 0 through 10 The only question remaining was in the clock group of period 11. .Janko's group was the sixth sporadic simple group to be whether the group of such matrices has found, and it was the firstto be discovered in more than 100 years. It is made up of all possi­ exactly 175,560 elements; if it did not, ble matrix products of the two generating matrices, such as AA, BB, ABA, BBAABABBB Janko's analysis would have yielded and so forth. There are exactly 23 X 3 X 5 X 7 X 11 X 19, or 175,560, matrices in the group. This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 107 All use ©subject 1985 SCIENTIFICto http://about.jstor.org/terms AMERICAN, INC Canada constructed the larger one, J3. new site, and by the time the rush was maticians have come to realize that The calculations for J3 were too com­ over four more sporadic groups had Janko's discovery of J2 and J3 was al­ plicated to be done by hand, and so been uncovered. The speed with which most miraculous. Out of a myriad of they were carried out on a computer. these early discoveries were made only potential variants from which to se­ The scene now shifts, for Hall and reinforced the impression that the lect, he had chosen one of the few pos­ Wales had been able to construct J2 as number of still undetected sporadic sibilities capable of flowering. a familiar kind of group of permuta­ groups was large-possibly even infi­ I have stressed that all finite groups tions. Group theorists flocked to the nite. With hindsight, however, mathe- can be broken down into simple com-

4

2 2 2 2

3 3 3

(12)(34)*( 123) =( 134)

4 4

3 3

cl� 1

4 4 2

3 3 3

ROTATIONS OF A REGULAR TETRAHEDRON that preserve its orientation in space form a group with 12 elements. The rotations are combined sequentially, that is, the group product of two rotations is the result of doing one rotation and then the second. The four rotations in the top row are the identity rotation, which does nothing, and the three "edge" 4 rotations. Each of the edge rotations interchanges two pairs of vertexes of the tetrahedron; 2 for example, the rotation (12)(34) interchanges the vertexes at positions 1 and 2 and the vertexes at positions 3 and 4. The middle row shows the group product of each edge rotation followed by the "face" rotation (123), which sends the vertex at position 1 to position 2, the vertex at position 2 to position 3 and the vertex at position 3 back to position 1. The bottom row shows the group product of each edge rotation followed by the face rotation (132). Each 3 product is equivalent to a face rotation. The rotations are applied to the tetrahedron at left.

108 This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC ponents. What I have not done so far, SECOND ELEMENT however, is to say exactly what a sim­ * (1 ) (12) (13) (14) (123) (134) (243) (142) (132) (234) (124) (143) ple group is or how the decomposition (34) (24) (23) is accomplished. Both points are re­ (12) (13) (14) lated to the mathematical procedure (1 ) (1 ) (34) (24) (24) (123) (134) (243) (142) (132) (234) (124) (143) called telescoping, which I mentioned briefly above. When a telescopic im­ (12) (12) (1 ) (14) (13) (134) (123) (142) (243) (23�) (132) (143) (124) age of a group is made, multiplica­ (34) (34) (23) (24) tion in the parent group G is reflected (13) (13) (14) (12) (243) (142) (123) (134) (124) (143) (132) (234) in the image group G', although nor­ (24) (24) (23) ('1) (34) mally in a reduced form. The process can be compared to observing an ob­ (14) (14) (13) (12) ( (23) (23) (24) (34) (1) (142) (243) (134) (123) (143) (124) 234) (132) ject through the reverse end of a tele­

scope: the general features of the ob­ t­ (13) (14) (12) Z (123) (123) (243) (142) (134) (132) (124) (143) (234) (1) ject are preserved but its apparent size W (24) (23) (34) is diminished. W:2; (12) (14) (13) In the mathematical process of tele­ --' (134) (134) (142) (243) (123) (234) (143) (124) (132) (1) W (34) (23) (24) scoping each element a of the group G � must be associated with an element a' a: (243) (243) (123) (134) (142) (124) (132) (234) (143) (13) (1 ) (12) (14) of the group G'; the element a' is called u::: (24) (34) (23) the image of a. Several elements of G (14) (12) (13) may have the same image in G', and (142) (142) (134) (123) (243) (143) (234) (132) (124) (23) (34) (1 ) (24) this accounts for the reduction in size. (14) (12) (13) In addition every element of G' must (132) (132) (143) (234) (124) (1 ) (23) (34) (24) (123) (142) (134) (243) be the image of at least one element of G. If such a way of associating the ele­ (234) (234) (124) (132) (143) (12) (13) (1) (14) (134) (243) (123) (142) ments of a group G with the elements (34) (24) (23) of a group G' is to be called telescop­ (13) (12) (14) ing, the group operations in Gand in G' (124) (124) (234) (143) (132) (24) (34) (23) (1 ) (243) (134) (142) (123) must be closely interrelated: if the ele­ ments a' and b' of the group G' are the (143) (143) (132) (124) (234) (14) (1 ) (13) (12) (142) (123) (243) (134) images of the elements a and b of the (23) (24) (34) group G, the product a' * b' in G' must be the image of the prod uct a * b in G. GROUP MULTIPLICATION TABLE is given for the tetrahedron group in the illustration An arbitrary group G always has at on the opposite page. Note that the group contains elements that do not commute. For ex­ least two telescopic images. One of ample, the rotation (1 2)(34) followed by the rotation (123) is equivalent to the rotation them is its mirror image: G' is just the (134), but if the order of the first two rotations is reversed, their group product is (243). The same group as Gand each element in G red region at the upper left of the table shows that the group product of any two edge rota­ is its own image. In effect, G is left tions of the tetrahedron is either an edge rotation or the identity rotation. The three edge ro­ untouched, and so the product condi­ tations and the identity rotation form a group by themselves. Such a group within a group is tion for telescoping is satisfied. The called a subgroup. The table can be divided into nine square regions, as is indicated by the color coding. Within each colored region only four group elements are listed: in the red other telescopic image of any group G squares only the elements in the subgroup of edge rotations appear; in the blue squares are is its point image: here G' is the group the group products of the four "red" elements with the face rotation (123), and in the green that has only one element, namely the squares are the group products of the four "red" elements with the face rotation (132). identity e', and e' is the image of ev­ ery element of G. Then no matter how the elements a and b are chosen in G, the three elements a, b and a * b all groups of prime period are all simple tain its spatial orientation: one set of ' have the same image e in G'. Because groups, and, because there are infinite­ three positions for each of the four e' * e' is equal to e' in G', the condi­ ly many primes, such groups form the faces. The 12 rotations in the group tion for telescoping is again satisfied. first of the 18 infinite families of finite, can be designated by the way in which simple groups. they interchange the vertexes of the he definition of a simple group can It will be useful to give an example tetrahedron. For example, the "edge" Tnow be given: it is a group whose of a nonsimple group to clarify how rotation that exchanges the vertex at only telescopic images are its mirror the process of telescoping decomposes position 1 with the vertex at position 2 and point images. The most elementa­ a group into its simple components. and the vertex at position 3 with the ry examples of simple groups are the The set of all the rotations of a regular vertex at positiori 4 can be written clock groups whose period is a prime tetrahedron that do not change the (12)(34); the "face" rotation that sends number p. Indeed, an elementary theo­ spatial orientation of the tetrahedron the vertex at position 1 to position 2, rem in group theory states that the forms a group. The result of two such the vertex at position 2 to position 3 number of elements in a telescopic im­ rotations can be expressed as another and the vertex at position 3 back to age must evenly divide the number of (possibly a third) kind of rotation, and position 1 can be written (123) [see il­ elements in the parent group. Since a for every possible rotation there is an lustration on opposite page]. prime number has no divisors other inverse rotation that brings the tetra­ One can now show that this group G than 1 and itself, any telescopic image hedron back to its starting position. of rotations of the regular tetrahedron of a clock group of prime period p can The "rotation" that does nothing to the has a telescopic image that is neither its contain only one or p elements. This tetrahedron serves as the identity ele­ mirror nor its point image. The exis­ means that the only telescopic images ment. There are 12 elements in this tence of such a telescopic image will of such clock groups are their mirror group, which correspond to the 12 po­ imply that the tetrahedron group is not and point images. Hence the clock sitions of the tetrahedron that main- simple. One takes the clock group of

This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 109 All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC SIMPLE-GROUP COMPONENT of the tetrahedron group is ob­ gmup. The illustration shows how the table for addition modulo 3 tained by finding a so-called telescopic image of the tetrahedron in the clock group reflects the multiplication table for the tetrahe­ group. Each element colored red in the multiplication table of the dron group. For example, the group product of any two "blue" rota­ tetrahedron group is associated with the element 0 in the clock tions is equivalent to a "green" rotation, just as the sum of 1 and 1, group of period 3. Each element colored blue in the table is asso­ the two "blue" elements in the clock group, is 2, the "green" element ciated with the element 1 in the clock group, and each element col­ in the clock group. Note that the telescopic image of every element ored green in the table is associated with the element 2 in the clock in the subgroup of edge rotations is 0, the identity in the clock group.

This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 110 All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC period 3 as the image group G'. The element of the clock group of period 2. It is easy to show that the clock tetrahedron group G can then be These two elements also form a group groups of prime period are the only mapped onto G' as follows: the identi­ by themselves, and the only telescopic commutative simple groups. In fact, if ty element 0 of G' is to be the image of images of a group with two elements a commutative group contains a com­ the four elements (1),(12)(34), (1 3)(24) art! its point image and its mirror im­ posite number n of elements, then for and (14)(23) of G; the element 1 of G' age. Hence the subgroup made up of every number d that divides n evenly, is to be the image of the four elements the edge rotation (12)(34) and the iden­ the group has a telescopic image con­ (123), (134), (243) and (142) of G, and tity is itself a simple group, equivalent taining exactly d elements. Since the the element 2 of G' is to be the image to the clock group of period 2. Since number d need not be equal to 1 or to of the four elements (132), (234), (124) no further telescoping is possible, the n, such a commutative group is not and (143) of G [see illustration on oppo­ tetrahedron group G must have three simple. For example, the group of the site page]. simple components, namely the clock four edge rotations of the tetrahedron Considerable checking is needed to groups of periods 3, 2 and 2. Since G is a commutative group for which n is verify that this way of associating contains 12 elements and 3 times 2 equal to 4. Since 2 divides 4, the num­ elements has the telescopic property. times 2 is 12, this result checks with ber d can be 2, and so the group has a For example, consider the elements my remark above that the product of telescopic image with two elements, as (123) and (134). Their group product the number of elements in the simple discussed above. Every simple group (123) * (134) in G, which is the result of components must be equal to the num­ except the clock groups therefore con­ applying first the rotation (123) and ber of elements in the parent group. tains at least one pair of elements a and then the rotation (134) to the tetrahe­ dron, is equal to the rotation (124) [see illustration on page 109]. These three ele­ ments have the images 1, 1 and 2 re­ spectively in G'. Because the group product 1 * 1 in G' is 1 + 1 (modulo 3), which is equal to the element 2 in G', the telescopic condition holds for the pair of elements (123) and (134). Since Gcontains 12 elements whereas G' has only three, G' is indeed neither the mir­ ror image nor the point image of G. Thus G is not simple.

ow then does one find the sim­ H ple components of the rotation group G? One of these components is the clock group G' of period 3. Indeed, G' is a telescopic image of G, and G' is simple since 3 is a prime number. The next step is to focus on all the elements of G whose image is the identity ele­ ment e' of G': the four rotations (1), (12)(34), (13)(24) and (14)(23). These four elements form a group by them­ selves; such a group within a group is called a subgroup. The group con­ sists of the four edge rotations that bring each of the four faces of the tet­ rahedron to its front but do not rotate a face once it has reached the front po­ sition [see illustration on page 108]. The subgroup of edge rotations is not simple either: it too has a telescop­ ic image in addition to its mirror and point images, namely the clock group of period 2. Here the identity element of the clock group is the image of the edge rotation (12)(34) and the identity element (1), and the second element of the clock group is the image of the edge rotations (13)(24) and (14)(23). TELESCOPIC IMAGE of the subgroup of edge rotations gives the second simple-group Since the clock group of period 2 is component of the tetrahedron group, The edge rotation (12)(34) and the identity rotation also simple, that group is a second sim­ (1) form a second, nested subgroup (color). They are associated with the identity element ple component in the decomposition o in the clock group of period 2. The other two elements in the subgroup of edge rotations, namely the rotations (13)(24) and (14)(23), are associated with the element 1 in the clock of the parent tetrahedron group G [see illustration at right]. group ( gray). Addition modulo 2 in the clock group reflects group multiplication in the subgroup of edge rotations. For example, the group product of a "colored" rotation and a The final step in the decomposition "gray" one is equivalent to a "gray" rotation, just as the sum modulo 2 of 0, the "colored" of G is to focus on the two elements element in the clock group, and 1, the "gray" element in the clock group, is equal to 1, the (12)(34) and (1) in the group of edge "gray" element in the clock group. Thus the clock group of period 2 is also a simple-group rotations whose image is the identity component. A second clock group of period 2 is the third such component (not shown).

This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 111 All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC b for which a * b is not equal to b * a. other hand, DCBA is out of order in six tions of degree less than or equal to 4 The smallest example of a noncom­ ways, and so it is an even permutation and the nature of the solutions of such mutative simple group contains 60 ele­ of the letters. The dodecahedron group equations of degree greater than or ments. It can be described as the group is structurally identical with the group equal to 5. of rotations of a regular dodecahedron of even permutations of five letters. that do not change the spatial orienta­ Although the tetrahedron group is The other families of simple groups tion of the dodecahedron The 60 ele­ not a simple group, all even permu­ are harder to describe. They all ments in the group correspond to the tations of fiveor more letters form have representations as groups of five possible positions for each of the simple, noncommutative groups. Such square matrices of suitable size. In 12 faces of the dodecahedron [see illus­ groups, which are called the alter­ some cases the groups are initially tration on page 105]. nating groups on n letters, make up defined by means of such matrices, The dodecahedron group is close­ the second infinite family of simple whereas in others considerable work is ly related to the rotation group of the groups. Incidentally, it is this distinc­ required to obtain their matrix repre­ tetrahedron. The tetrahedron group is tion between the alternating groups sentations. The same statement holds structurally identical with the group of of degree less than or equal to 4 and for the sporadic simple groups, but so-called even permutations of four the alternating groups of degree great­ there the situation is even more com­ letters. For example, if the letters er than or equal to 5 that underlies plicated: in some cases the matrix de­ ABCDare permuted to CBAD, the per­ the work of Galois on the theory of scriptions are based on computer cal­ mutation is an odd one because it is out equations. The distinction explains the culations. Remarkably, even though of order in three ways: C precedes B, C sharp differences between the nature the monster is the largest of the spo­ precedes A and B precedes A. On the of the solutions of polynomial equa- radic groups, Griess was able to find a representation for it entirely by hand. His construction is given in terms of square matrices of size 196,883 by 196,883 whose entries are taken from the complex numbers. The names and KNOWN . . SIMPLE GROUPS� sizes of all the families of simple .. groups as well as the sporadic groups . . . are listed in a table [see illustration on . . . . · . pages 114 and 115]. · . .. .

.. Many families of noncommutative

· . . simple groups had been discovered by · . . the turn of the century, and each such group as well as Mathieu's five sporad­

. ...

. ic groups contained an even number of ' . ' UNKNOWN .. . elements. This fact soon led to a nat­ . . SIMPLE GROUP . ural conjecture: every noncommuta­

' . . tive finite, simple group, known or un­ ...... known, must contain an even number . . . . : ' " . . : . . , . . . of elements. It was not until 1962,

. .. however, that the by then celebrated

. . conjecture was verified by Walter Feit . .

. and John Thompson, both then at the . .

. . University of Chicago. The complex­ .. . ity of the proof of this easily under­ stood statement foreshadowed the ex­ treme length of the complete classi­ fication of the simple groups. The BRAUER proof of the Feit-Thompson theorem HALFWAY LINE filled one entire issue of the Pacific Journal 0/ Mathematics, 255 pages in all. In 1965 Feit and Thompson were awarded the Cole Prize in Algebra for their work. Their theorem can also be viewed as SIMPLE GROUPS an example of a restricted classifica­ NOT RESEMBLING tion theorem. In this form it states that KNOWN GROUPS the only simple groups containing an PROOF OF CLASSIFICATION THEOREM for all finite, simple groups required show­ odd number of elements are the clock ing that any simple group, with an unknown but perhaps arbitrarily complicated subgroup groups whose period is an odd prime structure, has a subgroup structure equivalent to that of one of the known simple groups. number. Analogously the aim of the The strategy of the proof is represented by the illustration, where the known simple groups full classificationtheorem is to obtain are the dots at the boundary of the 'enclosed region. Toward the inside of the region the dots a list of all the finite, simple groups represent groups whose subgroup structures are increasingly far removed from the sub­ without any restriction on the number group structures of the known simple groups. At the center of the region is the arbitrary of elements they may contain. As I not­ simple group of unknown subgroup structure. Each stage in the proof of the theorem is rep­ ed above, the finaltheorem states that resented by the movement of the unknown simple group toward the boundary, that is, at each stage the subgroup structure of the arbitrary group is shown to bear a closer resem­ the complete list is made up of the 18 blance to one of the simple groups on the boundary, The work of Richard Brauer led to a regular, infinitefamilies of groups and criterion that describes the stage about halfway to the boundary. The extreme length of the the 26 sporadic groups. proof can be explained by the fact that some 100 paths to the boundary had to be explored. Since more than 250 pages were re-

This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 112 All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC quired to prove the Feit-Thompson Since every boundary point is a pos­ theorem, it is not surprising that a sible destination, the analysis inevita­ STIMULATE INQUIRING, proof of the much more complicated bly requires a large number of paths to YOUNG MINDS ... conclusion of the full classification be explored on the journey from the With a THINGS membership. Kits mailed theorem has taken many thousands of central point to the boundary. Distinct each month have materials to use. and pages. The length of the proof cannot paths arise at forks in the journey; the a booklet giving background on the be attributed to the difficulty of de­ forks correspond to stages at which the sCience subject and easy to 'follow In· struct,ons for excitmg demonstrations scribing the known simple groups. The internal structure of the given group and experi ments. clock groups can be described in a few can have several alternative shapes. In Designed for young people 10 to 16. lines, yet it took Feit and Thompson the end roughly 100 different paths parents often enjOy them too' several hundred pages to show that had to be followed before the proof there were no other simple groups con­ was completed. Thus the classification taining an odd number of elements. theorem was actually the sum of about The length of the classification theo­ 100 separate theorems, which taken rem must be traced to another source. together yield the desired conclusion. In the 1940's and 1950's the pioneer­ hat source lies in the complexity of ing work of Richard Brauer on cen­ Tthe subgroup structure of an arbi­ tralizers of involutions established one trary finite group compared with that of the most important milestones of a simple group-just as a complex along the journey to the boundary. molecule has a far more tangled inter­ Even though not all elements of a nal structure than a single atom does. group necessarily commute with one To appreciate the nature of the classifi­ another, it is useful to consider the set IDEAS FOR SCIENCE FAIRS'AN INTERESTING cation problem, imagine that the finite, of all the elements in a group that com­ HOBBY·A HELP IN DECIDING ON A CAREER· A simple groups are spread throughout a mute with a given element a, that is, Gin THArs FOR THE WHOLE YEAR. region of the plane [see illustration on the set of all elements g in the group for JOIN NOW' opposite page). which g * a is equal to a * g. This set is MAIL WITH REMITIANCE TO: Group-theorists could not begin called the centralizer of the element a. their proof with the assumption that It is easy to check that such centraliz­ THINGS of Science 819 WASHINGTON CROSSING RD. any simple group has a subgroup ers are themselves groups when their NEWTOWN. PA 18940 structure approximately the same as elements are combined according to GIFT CARD SENT IF REQUESTED that of a known simple group, for in the operation in the parent group, and CANADA AND MEXICO ADD $6.50: essence that is what the theorem sets so they are subgroups. A trivial exam­ OTHER COUNTRIES ADD $9.00. out to prove. Hence one obtains a ple is the centralizer of the identity ele­ more accurate picture of what had to ment e, which is always the entire par­ be proved if one allows the subgroup ent group since one of the definingcon­ structure of the simple groups in the ditions of a group is that every element planar region to vary over the same in the group must commute with the wide range of complexity as the struc­ identity element. A more substantive ture of all finite groups. The dots on example is the centralizer of the ele­ the boundary of the region represent ment (123) in the rotation group of the the known simple groups, the dots near tetrahedron: the centralizer of (123) the boundary represent groups whose consists of the identity, the element subgroup structures closely resemble (123) itself and the element (1 32). the structures of the groups corre­ sponding to the nearest dots on the rauer's idea was to focus on those boundary, and the dots deeper in the B group elements a other than the interior of the region represent groups identity element e for which the group whose structures bear progressively product a * a is equal to e. Group ele­ There's a lot worth less resemblance to the structures of ments having this property are called saving in this country. the known simple groups. involutions, and it is easy to show that At the outset of the proof, one is any group with an even number of ele­ Today more Americans given an entirely unknown simple ments contains involutions. In view who value the best of yester­ group whose internal structure, as far of the Feit-Thompson theorem, every day are saving and using old as anyone can tell, is as intricate as that noncommutative simple group there­ buildings, waterfront areas of any nons imp Ie group. Represent the fore contains involutions. Brauer be­ and even neighborhoods. given group with a dot in the middle of gan by calculating centralizers of in­ Preservation saves energy, the planar region. To prove the theo­ volutions in several of the 18 regular materials and the artistry of rem the given group must be forced, families and observed that they had these quality structures. through a series of mathematical ar­ the same general structure as the par­ Help preserve what's guments, to coincide with one of the ent group, but in an embryonic form. worth saving in your com­ points on the boundary. Each little He then wondered whether it might be munity. Contact the National move of the dot toward the boundary possible to reconstruct the entire par­ Trust, P.O. Box 2800, is to be understood as the result of a ent group solely from a knowledge of Washington, D.C. 20013. refinement in the initially unknown these centralizers, and in time he went subgroup structure of the given group. on to resolve the question affirmatively As the dot moves closer to the bounda­ in some important special cases. ry, the internal structure of the given Not only does the work of Brauer Nati�or group begins to take on the shape of underlie the discovery of many of the Historic Preservation one of the known simple groups. sporadic groups but also it provided an Preservation builds the natKln This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 113 All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC operational procedure for subdividing tion is just a small portion of the entire who took part in the finaldecade of the journey to the boundary of the pla­ group. The aim of the second stage of work on the classification proof, I nar region of simple groups into two the proof is to extend this local infor­ should mention the name of Michael parts. First, prove that the centralizer mation about centralizers of involu­ Aschbacher of Caltech. In 1972, in a of an involution in an unknown simple tions to a global equivalence, there­ series of lectures at the University of group closely resembles the centralizer by showing that the unknown simple Chicago, I presented a 16-step pro­ of an involution in one of the known group is the same as one of the known gram for determining all the finite, sim­ simple groups. The analyses forming simple groups. Paths near the bounda­ ple groups. I suggested, in effect,the the basis for paths near the central dot ry points are concerned with showing paths that would have to be followed are related to this aspect of the classifi­ how the known simple groups are de­ on the journey to the boundary. I pre­ cation problem. termined from information about the dicted a successful outcome by the end Knowing that the centralizer of an centralizers of their involutions. The of the century, but my grand design involution in the unknown group is the separation 'of the proof of the classifi­ and its accompanying optimism were same as the centralizer of an involu­ cation theorem into these two phases met with considerable skepticism. tion in a known group is not at all can be pictured as a closed curve inside Neither the audience nor I had reck­ equivalent to knowing that the un­ the planar region, about halfway from oned with Aschbacher. Fresh out of known group is the same as the known the central dot to the boundary. graduate school, he had just entered group: the centralizer of an involu- Among the many mathematicians the field, and from that moment he be-

INFINITE FAMILIES OF FINITE, SIMPLE GROUPS NAME OF FAMILY NUMBER OF ELEMENTS

CLOCK GROUPS Zp OF p, WHERE P IS ANY PRIME NUMBER = = = 5 PRIME PERIOD P EXAMPLES: IZ,I 2; IZ,I 3; IZsl ;

ALTERNATING GROUPS Altn, THE EVEN PERMUTATIONS 1/2 x 1 x 2 x ... x n > = x x x x x = OF n LETTERS, n 4 EXAMPLES I Altsl 1/2 1 2 3 4 5 60; IAlt,1 = 3 60; . CHEVALLEY GROUPS LINEAR CHEVALLEY GROUPS

> > )2 X x x X n+1- 1)/d, d = G.C.D.(n + 1,q-1) An(q), n 1 OR q 3 qn(n+' (q' - 1) (q'- 1) ... (q EXAMPLES: IA,(4)1 = 4" ('+')" X ( 4' +1-1)/[G.C.D.(1 +1,4-1)J = 4 x 15 /1 = 60 ; IA,(5)1 = 5 x ( 5'-1 )/[G.C.D.(1 + 1,5-1)J = 12 0/2 = 60; IA,(2)1 = 16 8; . SYMPLECTIC CHEVALLEY GROUPS

> ' X x x X = Cn(q), n 2 qn (q'-1) (q' - 1 ) ... (q'n-1)/d, d G.C.D.(2,q-1) EXAMPLE: IC ,(2)1 = 2" x (2 '-1) x (2 '-1) x (2" ' -1)/[G.C.D.(2,2 -1)J = 5 12 x 3 x 1 5 x 63/1 = 1,4 51,520 ORTHOGONAL CHEVALLEY GROUPS n > > ' X x x X = Bn(q), 2 OR q 2 qn (q'-1) (q'-1) ... (q'n-1)/d, d G.C.D.(2,q-1)

qn1n-1) X (q n-1) x (q' - 1 ) x (q '-1) x ... X (q 2(n -1)-1)/d, d = G.C .D.(4,qn- 1 ) EXCEPTIONAL CHEVALLEY GROUPS G,(q), q > 1

x x x x s q36 (q12-1) (q9+1) (q8-1) (q'-1) x (q +1) x (q' -1)/d, d = G.C.D.(3,q + 1 ) SUZUKI GROUPS 'B,(q), q = 2m, mODO, m > 1 q' x (q' + 1 ) x (q - 1 )

REE GROUPS q' x (q'+1 ) x (q -1) = > = 'G,(q), q 3m, mODO, m 1 EXAMPLE: I'G,(3')1 = 2 7' x (27'+ 1) x (2 7 - 1 ) 10,073,444,472;

= 'F,(q), q 2m, mODO q12 x (q '+1) x (q' - 1 ) x (q '+1) x (q -1) [FOR q = 2, 'F,(2)'J [i'F,(2)'I = ' /2 I'F,(2)1 = '12 x 2 12 x ( 2' + 1) x (2' - 1 ) x (2 '+ 1) x (2 -1) = 1 7,971,200J

COMPLETE LIST of finite, simple groups, together with the num­ at the University of Heidelberg, Donald Higman of the University ber of elements in each group, is given here. The table on this page of Michigan, Charles Sims of Rutgers University, Jack McLaughlin lists the 18 infinite families of simple groups, and the table on the of Michigan, Suzuki, Arunas Rudvalis of the University of Massa­ opposite page lists the 26 sporadic simple groups. The letters A chusetts at Amherst, Dieter Held of the University of Mainz, Rich­ through G designating the infinite families are derived from Lie ard Lyons and Michael O'Nan of Rutgers, John Conway of the Uni­ theory, which is named after Sophus Lie. The families are named versity of Cambridge, Bernd Fischer of the University of Bielefeld, for Claude Chevalley, Robert Steinberg of the University of Califor­ Koichiro Harada of Ohio State University, John Thompson, now at nia at Los Angeles, Michio Suzuki of the University of Illinois and Cambridge, and Robert Griess, Jr., of Michigan. Individual simple Rimhak Ree of the Universitr of British Columbia. The sporadic groups that are members of a given infinite family are designated simple groups are named for Emile Mathieu, Zvonimir Janko, now by numerical values of the variables n, p and q, where /I can be any This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 114 All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC came the driving force behind my pro­ of the first proof includes some false algorithms, mathematical logic, geom­ gram. In rapid succession he proved starts, inefficiencies and duplication. If etry and number theory. The monster one astonishing theorem after another. we are successful in our current work, group is now known to have deep con­ Although there were many other ma­ the second-generation proof will be nections, not yet fully unraveled, with jor contributors to this final assault, only one-fifth as long as the first, with a the theory of elliptic fu nctions. Aschbacher alone was responsible for commensurate improvement in con­ The impact of the classification out­ shrinking my projected 30-year time­ ceptual clarity. By mathematical stan­ side mathematics is less clear. There table to a mere 10 years. dards a 3,OOO-page proof will still be has, however, been some speculation a proof of enormous length. Given that Griess's friendly giant, the mon­ hat does the future hold? My the complexity of the problem, how­ ster group, may enter into the formula­ Wcurrent work, along with that of ever, a substantially shorter proof will tion of a possible unified field theory oL several co-workers, is devoted to con­ have to await the introduction of to­ elementary particles. But whatever the structing a second-generation proof tally new methods. ultimate applications, finite-groupthe­ of the classification theorem. The first The classification of the finite, simple orists have succeeded in solving the proof encompasses many early papers groups is likely to have broad mathe­ most central problem of their subject, written long before there was any matical ramifications. Already the re­ which existed implicitly from the mo­ overall strategy for finishing the proof. sult has been applied to such diverse ment Galois first added the concept of Inevitably, therefore, the development areas of mathematics as the theory of a group to mathematics.

SPORADIC SIMPLE GROUPS NAME OF GROUP NUMBER OF ELEMENTS MATHIEU GROUPS M __ . . " � ...... _ .. _ .. .� .. ... ___ 2��)(��=.z.� ?O_...... _ ...... _. � ...._ ...__ . .... __ .. _...... __ _ M 5,0 12 _.__ ..__ ._ .. __ . ___ ..36��� 1--=--9 �g_��_.. ._ ... _ .. _ . .... __.. . --_ .. - M" 2' x 3' x 5 x 7 x 11 = 443,520 ._ ...... _...... _ ...... _-_., . ...- .... �---.,-..�------.-.-� .. ---.....- ..-...... - ..-. .... --.... _. .." ...... __ ...... _ ...... _ . . " ...._ ....._-_ .... -.--_ ...... , -- '--� -- M" 1 1 9 �_�__ _ __?� �=-><_��?�1 � ?3---= CJ,.?gO, �0_ .. � ...__ . ______..... _ ... _ ....._ ..... _ M" 210 x 33 X 5 x 7 x 11 x 23 = 24 4,823,040 JANKO GROUPS 23 x 3 x 5 x 7 x 11 x 19 = 175,560 J, . _. c ...... ::.. .-'-_ .... _-'-.....--'-� ..... __ .. c.=:..c. ..._._ ..... __ ...._ .... __ ..... _ .... __ ....- ----... .----.. -.-... ---�- ... 1- 2' X 33 X 5' x 7 = 604,800 J, .. _ ...... _-, ..... _ ...... _ ...... _ ...... - J3 2' X 3' x 5 x 1 7 x 19 = 50,2 32,960 I·· 2" X 33 X 5 x 7 x 11' x 23 x 29 x 31 x 37 x 43 8.68 x 10" J, � X X X 1 = HIGMAN·SIMS GROUP HS 29 3' 53 7 x 1 44,352,000 McLAUGHLIN GROUP Me 2' X 36 X 53 X 7 x 11 = 898,128,000 X X 1 SUZUKI SPORADIC GROUP Suz 2" 3' 5' x 7 x 11 x 3 � 4.4 8 x 10" RUDVALIS GROUP Ru 2" X 33 X 53 X 7 x 1 3 x 29 � 1.46 x 10" X X X 1 HELD GROUP He 210 33 5' 73 x 7 = 4,0 30,387, 200 X X X 11 LYO NS GROUP Ly 2' 3' 5F 7x x 31 x 37 x 67 � 5.18 x 10'6 X X 11 19 1 1 O'NAN GROUP ON 29 3' x 5 73 x x x 3 � 4.6 x 10" CONWAY GROUPS

X X X 11 8 C, 2" 39 5' 7' x x 1 3 x 23 � 4. 16 x 101 - t�� X 36 X 53 X 7 x 11 x 23 4.23 x 10" -c, ..__ ...... _ ...... _- .- � C3 210 X 3' X 53 X 7 x 11 x 23 � 4.96 x 10" FISCHER GROUPS 2 X ... __ __ x 39 5 F" ...... 1 " ' x 7 x 11 x 1 3 � 6.46 x 10 " 218 X 3" X 5' x 7 x 11 x 1 3 x 1 7 x 23 4.09 x 1 F" � 0'..' .... __ .. __ __ ...... __ ... ___ ... __ __ .. - F24' 2" X 3'6 X 5' X 73 x 11 x 1 3 x 1 7 x 23 x 29 � 1. 26 x 10" HARADA GROUP F, 2" X 36 X 56 X 7 x 11 x 1 9 � 2.73 x 10" 19 1 THOMPSON GROUP F3 215 X 3' 0 X 53 X 7 ' x 1 3 x x 3 � 9.07 x 10 '6 FISCHER GROUP F, X X X 19 ("BABY MONSTER") 2" 3" 56 7' x 11 x 1 3 x 1 7 x x 23 x 31 x 47 � 4.15 x 1033 FISCHER-GRIESS GROUP F, X X X 11' X 1 19 ("MONSTER," "FRIENDLY GIANT" ) 2'6 3'0 5' 76 x 33 x 1 7 x x 23 x 29 x 31 x 41 x 47 x 59 x 71 � 8.08 x 1053

positive integer and p can be any prime number. The variable q which is called the rank of the family, is restricted to the values 2, 4, must be equal to the number of elements in a finite number system, 6, 7 and 8 in the exceptional Chevalley groups. The Ree and Suzu­ and one can show that exactly one finite number system can be con­ ki groups are definedonly for values of q equal to an odd power In structed for a set of q elements only if q is an integral power of a of 2 or 3. The number of elements in each group in a family is given prime number. Hence the family names designate simple groups by the algebraic expression or the number in the second column of only for values of q equal to 2, 22, 23, ... ,3, 32,33, ... , 5, 52, 53, ... each table. Here d is the greatest common divisor (G.C.D.) of the and so on, unless other exceptions are noted. For example, the lin­ two numbers or algebraic expressions that appear in parentheses ear Chevalley group A [(3) is excluded from the list by the condi­ immediately following the abbreviation "G.C.D." In the examples tions imposed on the values of II and q, (,ecause A 1(3) is the non­ that are evaluated in the first table the vertical bars enclosing the simple group of rotations of a rl'l!U"lr tetrahedron. The integer II, name of a group designate tl' e number of elements in the group. This content downloaded from 128.198.6.196 on Mon, 16 Oct 2017 21:46:00 UTC 115 All use© 1985 subject SCIENTIFIC to http://about.jstor.org/terms AMERICAN, INC