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Simple finite groups are analogous to atoms. They are the basic units of construction for other, larger things. Simple finite groups combine to form larger, more complicated finite groups. The Enormous Theorem organizes these groups the way the periodic table organizes the elements. It says that every simple finite group belongs to one of three families-or to a fourth family of wild outliers. The largest of these rogues, called the Monster, has more than 1053 elements and exists in 196,883 dimensions. (There is even a whole field of investigation called monsterology in which researchers search for signs of the beast in other areas of math and science.) The first finite simple groups were identified by 1830, and by the 1890s mathematicians had made new inroads into finding more of those building blocks. Theorists also began to suspect the groups could all be put together in a big list.
Mathematicians in the early 20th century laid the foundation for the Enormous Theorem, but the guts of the proof did not materialize until midcentury. Between 1950 and 1980-a period which mathematician Daniel Gorenstein of Rutgers University called the "Thirty Years' War" -- heavyweights pushed the field of group theory further than ever before, finding finite simple groups and grouping them together into families. These mathematicians wielded 200-page manuscripts like algebraic machetes, cutting away abstract weeds to reveal the deepest foundations of symmetry. (Freeman Dyson of the Institute for Advanced Study in Princeton, N.J., referred to the onslaught of discovery of strange, beautiful groups as a "magnificent zoo.")
Those were heady times: Richard Foote, then a graduate student at the University of Cambridge and now a professor at the University of Vermont, once sat in a dank office and witnessed two famous theorists -- John Thompson, now at the University of Florida, and John Conway, now at Princeton University -- hashing out the details of a particularly unwieldy group. "It was amazing, like two Titans with lightning going between their brains," Foote says. "They never seemed to be at a loss for some absolutely wonderful and totally off-the-wall techniques for doing something. It was breathtaking."
It was during these decades that two of the proof's biggest milestones occurred. In 1963 a theorem by mathematicians Walter Feit and John Thompson laid out a recipe for finding more simple finite groups. After that breakthrough, in 1972 Gorenstein laid out a 16-step plan for proving the Enormous Theorem -- a project that would, once and for all, put all the finite simple groups in their place. It involved bringing together all the known finite simple groups, finding the missing ones, putting all the pieces into appropriate categories and proving there could not be any others. It was big, ambitious, unruly and, some said, implausible.
THE MAN WITH THE PLAN
YET GORENSTEIN was a charismatic algebraist, and his vision energized a new group of mathematicians -- with ambitions neither simple nor finite -- who were eager to make their mark. "He was a larger than life personality," says Lyons, who is at Rutgers. "He was tremendously aggressive in the way he conceived of problems and conceived of solutions. And he was very persuasive in convincing other people to help him."
Solomon, who describes his first encounter with group theory as "love at first sight," met Gorenstein in 1970. The National Science Foundation was hosting a summer institute on group theory at Bowdoin College, and every week mathematical celebrities were invited to the campus to give a lecture. Solomon, who was then a graduate student, remembers Gorenstein's visit vividly. The mathematical celebrity, just arrived from his summer home on Martha's Vineyard, was electrifying in both appearance and message.
"I'd never seen a mathematician in hot-pink pants before," Solomon recalls.
In 1972, Solomon says, most mathematicians thought that the proof would not be done by the end of the 20th century. But within four years the end was in sight. Gorenstein largely credited the inspired methods and feverish pace of Aschbacher, who is a professor at the California Institute of Technology, for hastening the proof's completion.
One reason the proof is so huge is that it stipulates that its list of finite simple groups is complete. That means the list includes …
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