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The Enormous Theorem Author(S): Daniel Gorenstein Source: Scientific American, Vol. 253, No. 6 (December 1985), Pp. 104-115 Publ

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The Enormous Theorem Author(S): Daniel Gorenstein Source: Scientific American, Vol. 253, No. 6 (December 1985), Pp. 104-115 Publ 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 JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Scientific American, a division of Nature America, Inc. is collaborating with JSTOR to digitize, preserve and extend access to Scientific American 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 effort of 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 play a 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.
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