The classification of all finite simple groups, started in 1830, was completed when Robert Louis Griess, Jr., constructed “the Monster,” a group with more than 8 10⁵³ elements, by hand.

There are few more general and powerful concepts in mathematics than that of a group. Groups describe the properties of objects as varied as the real numbers and the integers, card shuffling, codes, crystals, elementary particles, and the Rubik’s Cube puzzle—indeed, any object with some symmetry. The fundamental building blocks of finite groups are called the simple finite groups, much as atoms are the basic units used in constructing molecules. So it was with some excitement that mathematicians completed the proof of the classification of all the finite simple groups: The theorem states that any finite simple group either belongs to one of eighteen infinite families or is one of twenty-six exceptions, called “sporadic” groups. The excitement reached a climax when Robert Louis Griess, Jr., constructed the final finite simple group, “the Monster,” so named for its size. Mathematics, “the Monster”[Monster]
Simple group theory
“Monster, the” (finite simple group)[Monster, the]
Sporadic groups
[kw]Griess Constructs “the Monster,” the Last Sporadic Group (Jan. 14, 1980)
[kw]Monster,” the Last Sporadic Group, Griess Constructs “the (Jan. 14, 1980)
[kw]Last Sporadic Group, Griess Constructs ”the Monster,” the (Jan. 14, 1980)
[kw]Sporadic Group, Griess Constructs “the Monster,” the Last (Jan. 14, 1980)
Mathematics, “the Monster”[Monster]
Simple group theory
“Monster, the” (finite simple group)[Monster, the]
Sporadic groups
[g]North America;Jan. 14, 1980: Griess Constructs “the Monster,” the Last Sporadic Group[04040]
[g]United States;Jan. 14, 1980: Griess Constructs “the Monster,” the Last Sporadic Group[04040]
[c]Mathematics;Jan. 14, 1980: Griess Constructs “the Monster,” the Last Sporadic Group[04040]
Griess, Robert Louis, Jr.
Thompson, John Griggs
Brauer, Richard
Chevalley, Claude
Gorenstein, Daniel

In proving the classification, more than ten thousand journal pages have been filled by more than one hundred mathematicians, mainly from the United States, England, and Germany, but also from Japan, Australia, and Canada. The majority of the proof was published in some five hundred articles between the late 1940’s and the early 1980’s, although efforts to solve the problem span the entire 176-plus-year history of group theory. Simple groups having been recognized as fundamental from the start, the problem of classifying them gradually evolved into a field complete with its own specialized techniques.

Group theory was originally invented by the French mathematical prodigy Évariste Galois Galois, Évariste in 1830 to answer an age-old question about the solutions to polynomial equations. He defined a group to be a set of elements that satisfy four properties based on ordinary arithmetic: closure, identity, inverse, and associativity. The group of numbers under addition is an infinite group; an example of a finite group is the set of six ways to rotate or flip an equilateral triangle onto itself. The inverse of one of these moves would be the move that reversed the original motion, and the identity is doing nothing at all.

Any finite group can be decomposed uniquely or factored into its component simple groups, just as any whole number can be uniquely decomposed into its prime factors. One can get an intuitive feel for what it means to factor a group by considering the above symmetry group of the equilateral triangle. The motions on the triangle can be naturally considered as a combination of flips and rotations. In this way, the symmetry group is factored into the two-member flipping group and the three-member rotation group. Because these factor groups cannot be factored any further, they are called simple. As it turns out, any group with a prime number of elements must be simple.

Although Galois had laid the foundation, it was not until the 1860’s that other mathematicians understood his work and built on it. In 1870, Camille Jordan Jordan, Camille established the existence of five additional infinite families of simple groups. One family consists of what are called the alternating groups, and the remaining four families are known as the classical simple groups. In 1889, Otto Hölder’s Hölder, Otto proof that any finite group has a unique decomposition spurred a systematic search for finite simple groups. By 1900, the list of simple groups with less than two thousand elements was completed. In addition, there were five “Mathieu” groups, discovered by Émile Mathieu Mathieu, Émile in 1861, that apparently did not fit into any of the established families and so were the first sporadic groups.

Meanwhile, in the period 1888-1894, Sophus Lie, Lie, Sophus Élie Cartan, Cartan, Élie and others had succeeded in classifying all of a certain type of infinite group called Lie groups. It became apparent that there was a connection between some of the simple Lie groups and the classical finite simple groups. In addition to those corresponding to the classical simple groups, there were five “exceptional Lie groups.” In 1901 and 1905, the American mathematician Leonard E. Dickson Dickson, Leonard E. was able to derive another infinite family of finite simple groups, the finite analog of one of the exceptional Lie groups. It was not until 1955 that the French mathematician Claude Chevalley constructed another finite group by clarifying the relationship between the Lie groups and their finite counterparts. Over the following six years, other mathematicians extended Chevalley’s analysis and uncovered the remainder of the sixteen families of “Lie type,” including ones with no strict analogy among the Lie groups. They are the only infinite families aside from those of prime order and the family of alternating groups.

The approach toward a proof of a complete classification proceeded on two flanks. First, there were attempts at restricted classification theorems, where all simple groups with a given property were enumerated. By making an exhaustive list of properties, the full theorem could be completed. Second, there were attempts to construct new sporadic groups from scratch. Though largely unfruitful, these efforts produced a few successes that provided insight for the categorical approach. An example of a restricted classification theorem is the celebrated “odd order theorem,” which states that all nonprime simple groups have an even number of elements. Originally conjectured by the English mathematician William Burnside Burnside, William in 1901, this theorem was not proven until 1963, by Walter Feit Feit, Walter and John Griggs Thompson. Their proof is significant for the tools developed as well as the result itself and so provided a great impetus for the classification. In particular, groups of even order must have an element, called an involution, which when combined with itself forms the identity. That nonprime simple groups must have an involution meant that the pioneering work of the German mathematician Richard Brauer on involutions would be central in characterizing simple groups.

In 1966, Zvonimir Janko Janko, Zvonimir found the first sporadic group in more than a century as a counterexample while trying to prove a certain restricted classification theorem. Janko’s example provided a theme for constructing other sporadic groups as exceptions to restricted classification theorems. With the recent classification of the infinite families, a new simple group came as a surprise to the mathematical community. As more sporadics were discovered—at a rate of one or two a year—it seemed possible that an effort at a full classification would be futile if there were an infinite number of sporadic groups. By 1972, however, a complete classification seemed conceivable, and the American mathematician Daniel Gorenstein presented a sixteen-step program in a series of lectures at the University of Chicago. The projected thirty-year time period to accomplish the program was shortened to less than ten years by the work of Michael Aschbacher, Aschbacher, Michael among many others. By the end of the decade, it was clear that no significant theoretical obstacles remained.

On January 14, 1980, Griess announced that he had succeeded in constructing the Monster, the largest and final sporadic group. Calculated by hand, the Monster has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements and contains most of the other sporadic groups embedded within it. A prior proposal for constructing the Monster by computer was estimated at a cost of $3 million for one year of computing power. The clinching stroke in the proof of the classification came, however, in January of 1981, when Simon Norton Norton, Simon demonstrated the uniqueness of the Monster group.

The classification was the culmination of the effort of a large part of modern finite group theory and represented a powerful new tool. After completion, there was some feeling that the field had worked itself out of a job; yet, as consequences of the theorem continued to develop, mathematicians realized that there was still much work ahead. Aside from implications for fields external to finite group theory, a fair portion of internal work remained because of the extraordinary length and complexity of the theorem. An organized effort toward simplification and “revisionism” was started among the team.

Applications outside the theory of finite groups include several theorems in related branches of mathematics, such as number theory, model theory, the theory of algorithms, and coding theory. The Mathieu groups were known previously to be related intimately to the theory of codes, and some of the other sporadic groups were first derived from their connection with self-correcting codes and the packing of spheres in very high dimensions. In general, questions in related fields that can be reduced to questions concerning all finite simple groups can be settled simply by checking through them, case by case. In comparison, the techniques developed were of limited use in other fields, which is perhaps surprising considering the fundamental nature of the theorem. Because of the structural similarity between finite simple groups and other mathematical structures, it is hoped that the general approach will provide guidance to the classification in these other cases. In addition, some of the intermediate results in proving the classification have been of external interest. The highly symmetrical Monster group, for example, appears unexpectedly in the coefficients of certain functions, though the precise connection has yet to be clearly elucidated.

A revision was necessary for several reasons. Questions arose concerning human verifiability of a proof that is more than ten thousand pages long and depended on long machine calculations. Some doubted that the proof was done correctly or optimally because it was so lengthy. It was surprising that so complex a taxonomy of simple groups could grow out of so simple a question. Careful examination of the proof, however, suggests that the complexity is inherent in the problem.

As time passes and as no new simple groups are discovered, confidence in the validity of the proof has grown. Moreover, the theorem evolved historically as a patchwork of many individual, sometimes redundant, sections. The program of revisionism involved culling together the essential parts of roughly a hundred articles and either tidying them up or rederiving them. Alternative methods of proof provided a check, especially if they depended less or not at all on prior results from the classification theorem. Even before the full theorem was complete, in 1970, Helmut Bender had introduced methods for revision, some of which were used in the later stages of the proof. The ongoing revision is completed in stages, and the increased unity and succinctness of approach is hoped to compress the proof to roughly three thousand pages. This “second generation proof” leaves a more structurally sound legacy for future mathematicians to benefit from the essential insights of the classification. Mathematics, “the Monster”[Monster]
Simple group theory
“Monster, the” (finite simple group)[Monster, the]
Sporadic groups

• Cartwright, Mark. “Ten Thousand Pages to Prove Simplicity.” New Scientist 105 (May 30, 1985): 26-30. Brief popular account traces the historical development of the classification as the triumph of the combined efforts of many key personages. Keeps discussion of mathematical concepts to a minimum.
• Gallian, Joseph A. “Finite Simple Groups.” In Contemporary Abstract Algebra. 5th ed. Boston: Houghton Mifflin, 2005. Chapter in a textbook for upperclass undergraduates surveys finite simple groups, using classification according to size, or the “range problem,” as a starting point. Includes exercises and a diverse introductory bibliography.
• _______. “The Search for Finite Simple Groups: The Eighty Year Quest for the Building Blocks of Group Theory Reflects Sporadic Growth Spurts Whenever New Basic Techniques Were Discovered.” Mathematics Magazine 49 (September, 1976): 163-180. Gallian has researched the history of group theory and has written several articles intended for college students. This article chronicles the history of finite simple groups and the range problem as would be appropriate for a student familiar with basic group theoretic concepts. Includes bibliography.
• Gorenstein, Daniel. “The Enormous Theorem.” Scientific American 253 (June, 1985): 104-115. Writing for the layperson, Gorenstein succeeds in explaining the nature of groups and simple groups in intuitive terms using plain examples. He also manages to allow glimpses into some of the processes that went into the classification and why it was such an important and complex endeavor. Includes a short, more technical, bibliography.
• Gorenstein, Daniel, Richard Lyons, and Ronald Solomon. The Classification of the Finite Simple Groups. Vols. 1-6. Providence, R.I.: American Mathematical Society, 1983-2005. One of the driving forces behind the classification, Gorenstein has taken a central role in its documentation and revision. Here lies the lengthy record for the nonspecialist professional mathematician. Some of the philosophy involved in the revision is included here.
• Hammond, Allen L. “Sporadic Groups: Exceptions, or Part of a Pattern?” Science 181 (July 13, 1973): 146-148. Brief exposition for the layperson about sporadic groups, their connection with error-correcting codes, and some of their uses.
• Hurley, James E, and Arunas Rudvalis. “Finite Simple Groups.” American Mathematical Monthly 84 (November, 1977): 693-714. Presents a mathematical sketch of the development of the classification theorem, intended for college mathematics students. Includes an extensive mathematical bibliography.

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