Algebraic groups consist of elements, such as the whole numbers (−3, −2, −1, 0, 1, 2, 3 …), and an operation. The operation—for example, addition—combines two elements. In order for the elements to form a group, it is necessary that the combination of two elements also belong to the group, that the order in which two successive operations are performed does not matter, that the group contain a neutral element, and that for each element there exists an inverse element. Whole numbers, therefore, form a group “under addition.” So do the even numbers, since the sum of two even numbers is also an even number and since the inverse value of, say, 4 is −4. In both of these cases the number 0 forms the neutral element, since any number plus 0 leaves the original number unchanged. Odd numbers, however, do not constitute a group under addition because the sum of two odd numbers is not an odd number.

Whole numbers and even numbers are groups that contain an infinite number of elements. But there are also small groups that only contain a finite number of elements. One example would be the “clock face group.” This group contains the numbers 1 to 12. If you choose the number 9 in this group and add 8 to it, the result displayed on the clock will be 5 (12 in this case is the neutral element, since adding 12 to any other number gives the same number).

One of the most significant mathematical achievements of the 20th century was the classification of all finite groups—an accomplishment that in terms of its importance can be compared to the decoding of DNA or to the development of a taxonomy for the animal kingdom by Carl von Linné in the 18th century. To accomplish this

truly mammoth task required the united efforts of dozens of mathematicians all over the world.

In 1982 the American mathematician Dan Gorenstein was able to declare victory in the battle for the classification of all finite groups. Gorenstein had been coordinating the worldwide efforts of group theorists. No fewer than 500 publications, comprising some 15,000 printed pages, were needed to prove that there exist exactly 18 families of finite simple groups and 26 groups of a different sort. Small wonder this theorem was dubbed the “enormous theorem.”

Back in the 1960s, most experts thought it would take until far into the 21st century before the work would be completed. Some very unusual groups had been discovered that could not be classified into any of the schemes that had so far been developed. They were called “sporadic simple groups”—sporadic because they were rare and simple because … no, there is nothing that could be considered simple in the usual sense.

At about that time, John Leech, a mathematician at Glasgow University in Scotland, was studying so-called high-dimensional lattices. A mathematical lattice can be envisioned as a wire mesh. The wire netting around a tennis court is a lattice in two dimensions. The climbing frame in a playground is a lattice in three dimensions. Three-dimensional lattices play a very important role in crystallography—for example, where they illustrate the physical arrangements of the atoms. But Leech was not satisfied with two or three dimensions. He had discovered a 24-dimensional lattice, that would, from then on, carry his name: the Leech lattice. He set about to investigate its properties.

The most important characteristic of a geometric body is its symmetry. Just as a symmetric die looks exactly the same after it has been turned around on any of its axes, a Leech lattice too can be twisted, turned, and flipped—albeit in 24 dimensions—and always remain similar to itself. If a body has more than one symmetry, it is possible to rotate it around one axis, then another one, then

around the first one but in the opposite direction, and so on. Since the body is symmetrical, it will look the same after each of the rotations. From this it follows that one can “add” rotations—by performing one after the other—without changing the perceived shape of the body. Furthermore, it is possible to reverse a rotation by turning the body around the axis in the other direction.

The facts that symmetries can be added and that to each rotation there exists an inverse rotation are precisely the requirements that define them as a group. (The neutral element is the “no rotation” rotation.) Hence, the rotations of a symmetrical body can be thought of as the elements of a group. The actual properties of the group depend on the particular body itself.

This is one of many instances in which different mathematical disciplines—in this case geometry and algebra—meet up. Henceforth, mathematicians could cope with geometric problems in the area of symmetry simply by making use of algebraic tools. Leech suspected that the symmetry group of his lattice was of considerable interest, but soon realized that he did not have the group theory skills necessary to analyze it. He tried to get others interested in the question but couldn't. Finally, he turned to a young colleague in Cambridge, John H. Conway.

Conway was the son of a school teacher and grew up in Liverpool. He was awarded his doctorate in Cambridge and was appointed lecturer in pure mathematics. Very soon, however, he started to suffer from depression and came close to a breakdown. He was unable to publish any research results. It was not that he doubted his competence, but how could he convince the world of his abilities unless he published? Leech’s lattice appeared at just the right time. It would turn out to be his life saver.

Conway was not a very well-to-do man and, to help supplement his meager income, the depressed mathematician had to take on students for private tuition. Understandably this left him with little time for research and hardly any time for his family. But the opportunity Leech offered him was the stepping stone for which the young Cambridge mathematician had long been hoping. He would

not let it pass him by. Over dinner one evening Conway explained to his wife that during the coming weeks he would be preoccupied with a very complex and very significant problem. He would be working on it every Wednesday from 6 p.m. until midnight and then again every Saturday from noon until midnight. But then, to Conway’s utter surprise, he solved the problem in no more than a single Saturday session. On that very afternoon Conway discovered that the group describing Leech’s lattice was nothing less than a hitherto undiscovered sporadic group.

The Conway group, as it was referred to from then on, contains a gigantic number of elements: 8,315,553,613, 086,720,000 to be exact. The mathematical community was taken completely by surprise by Conway’s breakthrough, which advanced worldwide efforts to classify finite groups a considerable distance. More importantly for Conway himself, the discovery boosted his self-confidence and transformed his mathematical career. He was elected a member of the Royal Society and has remained since then at the cutting edge of research in mathematics. In 1986 he accepted an appointment to Princeton University.

As an aside it should be noted that the Conway group is by far not the largest sporadic group. Still to come at the time was the so-called monster group, which was discovered in 1980 by Robert Griess from the University of Michigan. It contains close to 10⁵⁴ elements, thus possessing more elements than the universe contains particles. The monster group describes the symmetries of a lattice in 196,883-dimensional space. And then there is the so-called baby monster. Containing a “mere” 4 × 10³³ elements, it is still a few shoe sizes larger than the Conway group. Even mathematicians who do not easily lose their cool over curious objects find sporadic simple groups unusually bizarre.