The Classification Theorem for Finite Simple Groups

It has been a great century for mathematical groups of all shapes and sizes! They have been part and parcel of our daily lives for the last couple of millennia or more. Apart from ruling our bureaucracy, groups of transformations and symmetries are the keepers of law and order among mathematical structures, the sciences and the arts — as well as the sources of beauty. Every finite group admits a decomposition into a finite sequence of simple ones similar to the prime f`actorization of integers. But it has proven spectacularly difficult to obtain a systematic grasp of the set of all finite simple groups.

The first step was a 254 page proof in 1961 of a key conjecture dating back to the last century. Then a team of over 50 group theorists started snowballing under the leadership of Daniel Gorenstein who, in 1980, announced the result of their labors, recorded in some 15000 highly technical journal pages: every finite simple group belongs either to one of two fairly well understood infinite families or to one of 16 less tractable infinite lists or is one of the 26 so-called sporadic groups— a strange elusive lot whose discovery was a hunt full of hunches, surprises and insights, reminiscent of the chase for elementary particles. It was an extraordinarily fruitful proof. Within 2 years of its completion a conference was dedicated to applications in nearly all branches of mathematics. But beyond merely establishing truth a good proof must illuminate and explain. At first ascent everybody is just glad to have scaled the peak.

Then comes the search for a more elegant, easier, snappier route. By 1985 that second stage of the project was well under way. The "enormous proof" has set a new trend in mathematics. It is a True Tale of a Tower of Babel with a Happy Ending! All those mathematicians toiling side by side, if spread all over the globe, each with his own outlook, language, bag of tricks: constructing geometries, permuting objects, calculating characters, centralizing, fusing, signalizing and inventing all sorts of new terms for situations that had been lying there waiting to be recognized, named and used. They were not treading on each other's toes but collaborating in a prolific way unprecedented in the history of mathematics!

This wonderful happening did provoke lively professional discussions but not much attention from the popular media. It is difficult to advertise, unintelligible without technical explanations and lacking in historical romance. Much of the friendliness of the sporadic monster is lost on an audience gaping at its size, incapable of appreciating its capricious charms. And there are no melodramatic side shows; group theorists — especially finite ones — make up just about the sanest and nicest species among mathematicians. Glitter and glamour are not engendered by the laborious toil that went into the quest for a classification of the finite simple groups. Finally, the "second generation" has not yet completed its task, a good reason for holding back the popular fanfares.

But it is a great story in progress to be carried over into the new millennium!

VERENA HUBER-DYSON received her Ph.D. in mathematics at the University of Zuerich with a thesis in finite group theory, then, in 1948, came as a post doc. to the Institute for Advanced Study in Princeton. After a short but fruitful marriage she resumed her career, at UC Berkeley in the early sixties, then at the U of Illinois' at Chicago Circle and finally retired from the University of Calgary. Her research papers on the interface between Logic and Algebra concern decision problems in group theory. Her monograph Goedel's theorem: a workbook on formalization (Teubner Texte, Leipzig 1991) is an attempt at a self-contained interdisciplinary introduction to logic and the foundations of mathematics.