The Sporadic Groups

Almost all finite symmetry atoms (known technically as finite simple groups) fit into a ‘periodic table’. There are infinitely many entries in this table, and 26 exceptions — called sporadic groups — that do not fit in. Some readers of my book Symmetry and the Monster have asked why there are only a finite number of sporadic groups, rather than an infinite number, and the reason is that they all arise from quirks of one sort of another. These quirks disappear beyond a certain level, so the sporadic groups gradually peter out. Further down this page I give an example of a quirk that leads to the Mathieu group M24, and via that to the Leech lattice, and eventually the Monster.

In the mid-to-late nineteenth century, the French mathematician Émile Mathieu created five very exceptional groups of permutations, the largest being M24. Mathieu’s groups did not fit into the later periodic table, and remained the only exceptions for a hundred years. Then in 1966 the Croatian mathematician, Zvonimir Janko found a new one, now known as J1. 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 were the Fischer groups, which are vastly expanded versions of the three large Mathieu groups, along with Fischer’s discovery of a still larger exception now called the Baby Monster.

The final sporadic group that was discovered was due to Janko, and is known as J4. The fact that there are no more was not at all clear at the time, and was only established beyond reasonable doubt with the publication in 2004 of the missing piece in the Classification, namely the full analysis of the quasi-thin case.

A Quirk

As promised above, here is an example of a quirk. It involves permutations of six objects, and although it does not extend to larger numbers of objects, it does lead to a sequence of exceptional groups, ending in the Monster. Take a collection of n beads and consider the group of all permutations of these beads. This is called the symmetric group of degree n, and I’ll denote it by Symm(n). Its size is n! (n factorial), meaning 1×2×3×4×…×n. For example Symm(4) has size 24, Symm(5) has size 120, Symm(6) has size 720, and so on. In each case the subgroup of Symm(n) fixing one of the n beads is a copy of Symm(n-1). With n beads, Symm(n) must contain n copies of Symm(n-1), one for each bead, and there are no more, except when n is 6. In that case, Symm(6) contains 12 copies of Symm(5), double the number you would expect. This is extraordinary.

It means that the symmetric group of degree 6 can operate in two entirely different ways on a set of six beads. This allows Symm(6) to operate simultaneously on two sets of six beads in two different ways; the subgroup fixing a bead in one set does not fix any beads in the other set, and vice versa. This quirky property, which can only happen when n is 6, has interesting consequences. The operation of Symm(6) on the twelve beads extends to the Mathieu group M12, which permutes the beads among themselves, mingling one set of six with the other.

Moreover M12 has the same unusual property as Symm(6); it can operate on a set of twelve beads in two distinct ways. When M12 operates simultaneously on two sets of twelve beads in two different ways a similar procedure leads to the largest Mathieu group M24. But that is the end of the phenomenon; M24 can operate in only one way on 24 beads, so the trick cannot be repeated.

However, M24 can be expanded in a different direction, and can be used to generate the Leech lattice in 24 dimensions. This in turn leads to the Monster, but there the process stops. The Monster does not lead to any further sporadic groups. Quirky phenomena lead to other quirky phenomena, but it only goes so far, and after that there is nothing more.