
An enormous theorem: The classification of finite simple groups - msvan
http://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups
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moomin
A mathematician friend of mine once tried to explain some stuff about his
field. It started with homomorphisms on Riemann surfaces and then got well
beyond my comprehension. Anyway, after he'd been talking for about five
minutes he said:

"And this has the cardinality of the monster, and no-one knows why."

Mathematicians keep finding peculiar and deep relationships between their
subfields that, for the most part, are not yet understood. As the article
states, a proof isn't really complete until it's comprehensible.

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kartikkumar
Blows my mind that the proof shows something as tangible as 18 types + all the
sporadics. Since I have absolutely no insight into the field, does anyone know
if the 18 is special in any sense?

The difference to me between physics and maths is that the latter has the
capacity to be fully deducible from the axioms, and in that sense, it's
fascinating that somewhere buried in our system of numbers, groups, geometry
etc. lies a set of characteristics dictating the existence of those 18 types.

~~~
pavpanchekha
Abstract algebra has a tendency to reveal integers without explanation.
Consider for example the exceptional Lie groups, which is an analogous
problem. Lie groups, which characterize _continuous_ groups (as in, groups
with a geometry of sorts among the elements of the group), can be broken down
into simple Lie groups which can then be classified into 4 infinite families
and 5 exceptions.

Why five? Well, it is the number of graphs with a certain type of property
(Dynkin diagrams). And beyond this, I have no good answer: it just simply is
the number. You can write them all down, you can verify that there are no more
(this part is hard), but I do not know of any deep insight as to why.

I should note that the 26 and the 18 for finite simple groups were not just
computed numbers, either. A few attempts at a complete classification were
made and failed because they were missing a few cases. If I recall, it was
unclear for a time if the Monster group really was the largest sporadic group.

I also recall a story I was once told by Andrew Sutherland. He was working on
what is called a local-global principle, which would show that if a property
is true of elliptic curves at every prime, it is true over the rationals (with
some constraints). He found, after several failed proof attempts, that
something special happened at the prime 7. Why 7? Well, it is not too large
and not too small...

A last connection: consider the deterministic linear-time median-finding
algorithm. It begins: "break up the elements into groups of 5." In this case,
the reason 5 is the size of the groups is simply that the answer must be odd;
that 3 doesn't work; and that smaller is better.

Small numbers are weird.

~~~
kartikkumar
> Small numbers are weird.

Weird and wonderful :). Thanks for all the examples, makes me wanna crack open
my abstract algebra book!

I decided to go into engineering when I got out of high school, but part of me
was leaning towards pure maths. Still feel that yearning when I read things
like this. So incredibly fascinating, simple, and yet wildly complex at the
same time.

I've worked a lot on chaos theory in the context of celestial mechanics and
the gravitational 3-body problem (Newtonian). After having done my MSc thesis
in the field and nearing the completion of my PhD in it, I still can't get my
head around how such an incredibly simple problem can spawn such a wealth of
complexity and beauty. Kinda envious of Poincaré when he first mapped out the
tremendous complexity and structure underpinning it all.

Small numbers rule.

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beder
This is an incredibly interesting area of mathematics, and it's disappointing
that it's dying out. When I was an undergrad, I wanted to go into this field,
but I was dissuaded by my advisor because of the direction it was going (fewer
active researchers means fewer PhD positions, and ultimately, fewer academic
jobs).

I'm glad that there's an effort to consolidate and simplify the proof, since
as they say, it could end up effectively lost forever.

~~~
thaumasiotes
> I wanted to go into this field, but I was dissuaded by my advisor because of
> the direction it was going (fewer active researchers means fewer PhD
> positions, and ultimately, fewer academic jobs)

I don't understand the logic here. I've been told many times that the
academic's dream is to find a field where he's the _only_ active researcher.
Fewer active researchers means fewer PhD positions, sure, but it also means
more prominence for those who remain, less effort involved in finding
publishable results, etc.

My go-to example of offensively low-hanging fruit is De Morgan's law(s), which
I still can't believe were named after a person. They state, in plain english:

1\. If it is not the case that a collection of claims are all true, then one
or more of the claims is false.

2\. If it is not the case that any of a collection of claims is true, then
they are all false.

When you're the only active researcher in a field, you can have observations
like that named after you too!

~~~
bo1024
There's a big difference between being one of the only ones in a field because
it does not yet exist (i.e. you are founding the field) and because it is
exhausted (i.e. all the big results are already proven).

~~~
thaumasiotes
It is generally not the case in any area of math that "all the big results are
already proven".

~~~
bo1024
True. But, well, for instance, all the finite simple groups are classified.

(Not in the sense of the NSA.)

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Strilanc
Computer proofs of the classification of finite simple groups are also being
worked on.

Well, parts of it at least [1].

1: [http://research.microsoft.com/en-
us/news/features/gonthierpr...](http://research.microsoft.com/en-
us/news/features/gonthierproof-101112.aspx)

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CurtMonash
Glad to see a shout-out for Ronald Solomon. His group theory course was
probably my favorite math class. That's more about the material than him; we
just worked our way through the first part of Rotman. But he sure didn't do
anything to wreck it.

E.g., there was the time before class started that for some reason I went to
board and led the team in classifying finite groups up to order 60. He just
paused at the door when he saw that, smiled, and didn't start class until we
were done.

(Note: The reason that could be done in a few minutes is that for the purposes
of the exercise, prime numbers are trivial, and so are integers that are the
product of two distinct primes. That didn't leave a lot of other cases to
worry about.)

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gamegoblin

      The second advantage is power: if you have proved 
      something about regular polyhedra, then what you 
      have proved automatically holds true for every 
      polyhedron, whether it's a cube, a tetrahedron, or 
      some polyhedron that you have never even heard about.
    

Is this worded correctly/true? If I prove something is true for regular
polyhedra, then I don't believe that that extends to all polyhedra since
regular polyhedra are a subset of all polyhedra...

~~~
bo1024
It probably just means to say "holds true for every regular polyhedron".

------
ars
Are finite simple groups specifically about polyhedra, or was that just used
as an example?

~~~
ColinWright
The groups of symmetries of a finite object is always a finite group, and
every finite group arises as such. However, not all of the objects are
polyhedra, and in particular, not all of the objects are in 3 dimensions.

The monster is, for example, the symmetry group of an object in 196884
dimensions. It's not really a polyhedron.

