- Ronan, Mark.
Symmetry and the Monster.
Oxford: Oxford University Press, 2006.
ISBN 0-19-280722-6.
-
On the morning of May 30th, 1832, self-taught mathematical genius and
revolutionary firebrand
Évariste Galois
died in a duel in Paris, the reasons for which are forgotten; he was twenty
years old. The night before, he wrote a letter in which he urged that
his uncompleted mathematical work be sent to the preeminent
contemporary mathematicians Jacobi and Gauss; neither, however, ever
saw it. The work in question laid the foundations for
group theory,
an active area of mathematical research a century and three
quarters hence, and a cornerstone of the most fundamental
theories of physics: Noether's
Theorem demonstrates that conservation laws and physical
symmetries are two aspects of the same thing.
Finite groups,
which govern symmetries among a finite number of
discrete items (as opposed to, say, the rotations of a sphere,
which are continuously valued), can be arbitrarily complicated,
but, as shown by Galois, can be decomposed into one or more
simple
groups whose only normal subgroups are
the trivial subgroup of order one and the improper subgroup
consisting of the entire group itself: these
are the fundamental kinds of symmetries or, as this book
refers to them, the “atoms of symmetry”, and
there are only five categories (four of the five
categories are themselves infinite). The fifth category
are the sporadic
groups, which do not fit into any of the other categories.
The first was discovered by Émile Mathieu in 1861, and
between then and 1873 he found four more. As group theory continued
to develop, mathematicians kept finding more and more
of these sporadic groups, and nobody knew whether there were
only a finite number or infinitely many of them…until
recently.
Most research papers in mathematics are short and concise.
Some group theory papers are the exception, with two hundred
pagers packed with dense notation not uncommon. The
classification
theorem of finite groups is the ultimate outlier; it has
been likened to the Manhattan Project of pure mathematics. Consisting of
hundreds of papers published over decades by a large collection of
authors, it is estimated, if every component involved in the proof were
collected together, to be on the order of fifteen thousand
pages, many of which are so technical those not involved in the work
itself have extreme difficulty understanding them. (In fact, a
“Revision project” is currently underway with the goal
of restating the proof in a form which future generations
of mathematicians will be able to comprehend.) The last part of the
classification theorem, itself more than a thousand pages in length,
was not put into place until November 2004, so only then could one
say with complete confidence that there were only 26 sporadic
groups, all of which are known.
While these groups are “simple” in the sense of not being
able to be decomposed, the symmetries most of them represent are
of mind-boggling complexity. The
order of a
finite group is the number of elements it contains; for example,
the group of permutations on five items has an order of 5! = 120. The
simplest sporadic group has an order of 7920 and the biggest, well,
it's a monster. In fact, that's what it's called, the
“monster
group”, and its order is (deep breath):
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 =
246×320×59×76×112×133×17×19×23×29×31×41×47×59×71
If it helps, you can think of the monster as the group of rotations
in a space of 196,884 dimensions—much easier to
visualise, isn't it? In any case, that's how Robert Griess first
constructed the monster in 1982, in a 102 page paper
done without a computer.
In one of those “take your breath away” connections between
distant and apparently unrelated fields of mathematics, the divisors of
the order of the monster are precisely the 15
supersingular
primes, which are intimately related to the
j-function
of number theory. Other striking coincidences, or maybe
deep connections, link the monster group to the Lorentzian
geometry of general relativity, the multidimensional space of
string theory, and the enigmatic properties of the
number 163 in
number theory. In 1983, Freeman Dyson mused, “I have a
sneaking hope, a hope unsupported by any facts or any evidence, that
sometime in the twenty-first century physicists will stumble upon
the Monster group, built in some unsuspected way into the
structure of the universe.” Hey, stranger things
have happened.
This book, by a professional mathematician who is also a talented
populariser of the subject, tells the story of this quest. During his
career, he personally knew almost all of the people involved in the
classification project, and leavens the technical details with
biographical accounts and anecdotes of the protagonists. To avoid
potentially confusing mathematical jargon, he uses his own
nomenclature: “atom of symmetry” instead of
finite simple group, “deconstruction” instead of
decomposition, and so on. This sometimes creates its own
confusion, since the extended quotes from mathematicians use
the standard terminology; the reader should refer to the glossary
at the end of the book to resolve any such puzzlement.
November 2006 