Mark Ronan's website
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, but there are just 26 exceptions that do not fit in—they are called sporadic groups. 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 on 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 three 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.
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. And 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.