Anybody heard of the Monster Group? Did you realize that it is the head of the happy family? Like all families not everything is perfect and the happy family has outcasts.
Now what are we talking about? A new soap or reality TV series? Another music track when numbers of celebrity pop icons get a couple of bars to sing? Well, no. These terms refer to one of the less accessible achievements of twentieth century mathematics. It is known as the enormous theorem. Enormous, because the proof is estimated to run to over fifteen thousand pages. That’s like the complete Harry Potter series about six times over.
I’d better issue a health warning at this point. Anyone looking for one of my usual faintly-humorous-with-a-whiff-of-the-ridiculous pieces is going to be disappointed. This, as they say, is something completely different and is an attempt to make something abstract understandable or at least a little more accessible. And it’s much too big for a blog really, I just can’t think of anywhere else to put it. With that out of the way, we’ll press on then.
Since the enormous theorem is so big, I think we better approach it in easy stages.
Let’s start with clock arithmetic, of the hours. When we add a couple of positive numbers together in ordinary everyday arithmetic, the answer is always bigger than either of them, but in the clock version, that is not so. Three hours after 10 o’clock is 1 o’clock. The rule is simple, if the answer is more than twelve o’clock, like forty-five o’clock, you just keep taking twelve away until you come to a proper time, so that forty-five hours after mid-day is actually nine o’clock. Are we all happy so far? Well here’s another thing, there’s no multiplication in clock arithmetic, only addition, which makes clock arithmetic particularly simple to do, there are only twelve numbers and only addition (yes, I know, but subtraction is really just a special sort of addition). A construct like clock arithmetic with a number of elements and a single way of combining them is called a (finite) Group.
We are going to construct another finite group. Imagine a square. Turn it through 90 degrees and what you get looks the same as the square you started with, flip it over, same result, rotate it 180 degrees round a diagonal, same thing. Now let’s number the corners of the square anti-clockwise from one to four and do the same things: we can flip it, rotate it, turn it, always ending up with the square landing back on itself of course, but now we have a better insight into what we did, as the numbers on each corner have changed. For example if we turn it anti-clockwise through 90 degrees, 1 is where 2 was, 2 is where 3 was and so on. Make your own square, if you like, and convince yourself that there are eight and only eight possible positions of the corners. But there’s something a bit odd. If we flip and turn 90 degrees, we don’t get the same result as when we turn 90 degrees and then flip, so the order in which these operations are done is important because different orders sometimes give different results. And this is fundamentally different from what happens when we combine numbers in everyday and in hour clock arithmetic, for then the order of doing things doesn’t matter; two plus three is the same as three plus two and any other two numbers that you can think of behave the same way. If order is not important then the operation is described as commutative.
A Group is a broad church and includes those constructs where order does not, as well as where it does, matter. A group operation can be commutative or not.
Let’s think about the clock again and this time, think about the minute hand which gives a different clock arithmetic, this time with sixty as its base. Embedded within this clock arithmetic is the idea of quarters. These happen at fifteen minutes or a quarter past the hour, half past, quarter to and on the hour. But the quarters form another group, this time with only four elements. Quarter past, plus another half (two quarters) make up quarter to, and so on. This idea of a smaller group happily existing inside a bigger group is called a sub-group. There are sub-groups inside our square changing group too, one of them consists of the rotations only. Can you see what others there are (there are in fact two)? Now our clock arithmetic sub-group is a special sort of sub-group. We can collapse the whole base sixty clock arithmetic group onto it, we can talk of up-to-quarter-past, between-quarter-past-and half-past and so on. We’ve talked about two sorts of clock arithmetic, the 12 hour clock arithmetic and the 60 minute clock arithmetic. Both have the quarters as a collapsing sub-group. For some reason, mathematicians have coined the term “normal” for such a sub-group.
We have constructed groups above which have 4, 8, 12, and 60 elements. Mathematicians began studying groups in the eighteenth century. The first question that they began addressing themselves to was “if I give you a number, is there at least one group with that number of elements?” That’s an easy question to answer and it is “yes”; just construct a new clock-like arithmetic with the given number as the base. The second question was “if I give you a number, how many different groups can you construct with that number of elements?” That, as they say, is a horse of a very different colour and eventually metamorphosed into “can we classify groups in some way to make answering that second question a bit easier”. A French mathematician by the name of Evariste Galois noticed that certain groups had sub-groups that were not normal, and indeed that some had no normal sub-groups at all. Groups of this last type, devoid of normal sub-groups, are called “simple”. If the number of elements in the simple group is a prime number, then the name is descriptive, but if not, then … wow, simple they are not. For those who want a bit of technical spice, the smallest is the group of symmetries of the dodecahedron, and it also has sixty elements.
It turns out that simple groups bear a relation to finite groups similar to the one that prime numbers do to everyday arithmetic. All groups can be decomposed into simple groups, a bit like all numbers can be factored into primes. So, if we could classify all simple groups, then we would have at least classified the building blocks of groups.
And that is what the enormous theorem does, it describes all simple groups and it does so in a constructive manner showing how to make them. It says, roughly, that there are only four (or eighteen, depending on how big you let them be) different families of simple groups, two of which were known to our friend Galois who discovered the first simple group with a non-prime number of elements – it’s the one mentioned above, the dodecahedron’s symmetries. All the families have an infinite number of members. But, and this is where we came in, there are twenty six additional simple groups, including the so-called monster group, and nineteen of the others are (non-collapsing of course) sub-groups of the monster group. This collection of twenty is known as the happy family, leaving six outcastes or pariahs. The monster group is really, really big – it has more elements than the number of atoms that make up the Sun.
Now why is this important, you may ask. Well, although they started to be looked at to help understand algebraic equations, groups crop up all over the place, for example in music, chemistry and physics, so understanding groups means that we get a better understanding of the situations where they can be applied.
But what is unusual about this endeavour is the number of people involved. Mathematics used to be a solitary occupation, but it is becoming more and more a group (sorry, no pun intended) effort, over one hundred different men and women having contributed to the final result known as the enormous theorem.
Perhaps that’s why the imagery in the name is a family one, and the monster group’s other name is “the friendly giant”.
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