Symmetry: from Galois to the discovery of the Monster and Moonshine

The mathematical study of symmetry is called group theory. This is because the symmetry operations on an object, or the symmetries that preserve a particular pattern, form a group in the mathematical sense. One symmetry operation followed by another gives a third one in the same group, and this group embodies, in an abstract way, the symmetry of the object or pattern concerned. The application of groups to serious mathematical problems first arose in the work of …variste Galois, a young French mathematician who died after being fatally wounded in a duel at the age of twenty.

Group Theory

Mathematicians study groups in various ways, one of which is to deconstruct them into simpler groups. Those that cannot be deconstructed further — the very 'atoms' of the subject — are called 'simple' groups, though they can be very complicated. In the book, these finite simple groups are called 'atoms of symmetry', and the first ones were discovered by Galois in about 1830.

Finite Simple Groups

Most finite simple groups fit into a table, rather like the periodic table of chemical elements. Those in the table are called groups of Lie type, the term "Lie" (pronounced Lee) being in honour of the Norwegian mathematician Sophus Lie. His work in the late nineteenth century led to continuous groups — called Lie groups — and these in turn led to finite groups of 'Lie type'. The table of all such groups was complete by the early 1960s, but there were exceptions that did not fit in. They are called sporadic groups.

Sporadic Groups

In the mid-to-late nineteenth century, the French mathematician …mile Mathieu created five very exceptional groups of permutations, the largest of which is called M24. Mathieu's groups did not fit into the later periodic table, and remained the only exceptions for a hundred years, until the Croatian mathematician, Zvonimir Janko found a new one that he published in 1966. This inspired the search for other sporadic groups, and their discovery is an intriguing story involving a variety of methods: some geometric, some involving patterns exhibiting interesting permutations, and some by analyzing possible cross-sections (called 'involution centralizers' in group theory). These latter cases were very technical, and the construction of the sporadic group was a tricky business, usually involving computer techniques. The Monster — the largest sporadic group — was predicted by the cross-section method, but its size and complicated structure rendered computer methods impractical, and it had to be constructed by hand. There are two main threads that led to the Monster. One was the Leech Lattice and the Conway groups; the other was the Baby Monster discovered by Bernd Fischer.

The Leech Lattice

This is a 24-dimensional lattice created in the 1960s by John Leech in Scotland. He used a design discovered in the mid-1930s by the German mathematician, Ernst Witt who had created it in order to construct the largest Mathieu group M24. Leech used it to obtain a remarkable way of packing 24-dimensional spheres, a fact with useful applications to technology. The symmetry of this lattice was investigated in detail by the English mathematician, John Conway. It yielded several sporadic groups, including three new ones, now known as the Conway groups.

The Monster

The other thread that led to the Monster emerged from work of the German mathematician, Bernd Fischer who created three large and remarkable sporadic groups that are related to, but far larger than, the three largest Mathieu groups. Fischer then found a huge fourth one, later named the Baby Monster, and the Monster was predicted as an even larger sporadic group having the Baby as a cross-section. Fischer, in collaboration with Donald Livingstone and Michael Thorne in England, calculated the character table of the Monster (a square array of numbers giving immense information about the group in question). They assumed the Monster could operate in 196,883 dimensions — at minimum — a number calculated by Simon Norton at Cambridge. Norton worked out that the Monster, if it existed, would have to preserve an algebra structure in 196,884 dimensions, and the American mathematician, Robert Griess constructed it on that basis. His work used the Leech lattice, and his algebra structure yielded the Monster as its group of symmetries.

196,884

This number was the subject of a remarkable coincidence, first noticed in 1978 by the English mathematician John McKay, working in Canada. It is the first non-trivial dimension of a space the Monster operates in, and is also the first non-trivial coefficient of something called the j-function, which has miraculous properties in number theory (the study of whole numbers — a quite different branch of mathematics). McKay communicated this coincidence to the American mathematician John Thompson, who had a chair at Cambridge in England, but was visiting Princeton at the time. Thompson worked out other coincidences connecting the Monster with the j-function, and on returning to Cambridge he explained his findings to John Conway, who took up the matter in detail. Using the Monster's character table, which had been very recently constructed, he and Simon Norton proved that there was a definite connection between the Monster and the j-function, and dubbed the whole thing Moonshine (referring to a great mystery, as yet barely understood).

Moonshine

In 1979 Conway and Norton published a paper with the title Monstrous Moonshine, proving there really was a connection. They also proposed there should be a space in infinitely many dimensions, exhibiting connections between the Monster, the j‑function, and other j-functions. Their ideas were taken up by Igor Frenkel, James Lepowsky and Arne Meurman who constructed a suitable space they called the Moonshine Module, and it turned out to be connected to the mathematical physics of string theory. The Conway-Norton conjectures for the Moonshine module were later proved by the English mathematician, Richard Borcherds, who also gave another approach to the Monster by starting with a lattice in 26-dimensional space-time.

The Classification

All finite simple groups (i.e., finite symmetry atoms) have been found: each one is either a group of Lie type, or one of 26 sporadic groups. The proof of this fact — called the Classification — was a major program of mathematics research that got seriously underway in the early 1960s. The results appeared to be nearly complete by about 1980, but the details were extremely technical, and were scattered through a large number of research papers, numbering about 10,000 printed pages. A complete revision of the Classification has been underway for twenty years, and is still continuing. Moreover there was one feature of the Classification — the quasi-thin case — that had never been published, but this was finally settled in 2004.