An enormous theorem: The classification of finite simple groups(plus.maths.org)
plus.maths.org
An enormous theorem: The classification of finite simple groups
http://plus.maths.org/content/enormous-theorem-classification-finite-simple-groups
7 comments
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.
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.
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.
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.
> 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.
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.
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.
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.
> 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!
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!
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).
It is generally not the case in any area of math that "all the big results are already proven".
True. But, well, for instance, all the finite simple groups are classified.
(Not in the sense of the NSA.)
(Not in the sense of the NSA.)
We have a professor who is one of a few dozen active researchers in his field, and I have developed some algorithms for him. It is really trivial to publish in fields like that. And back in the 50s and 60s anybody could get an algorithm named after them.
Sorry to let you know, but most professors are in fields like that. That's why I left the subject. :-(
Really? Most of our CS professors are in much more well-established fields. Our department head is in search engines, my advisor is data mining, we have some GPU computing folks, and some database security guys, etc. Then the one guy in a highly esoteric field, where a "conference" involves a half dozen people meeting for a week.
I was in math.
I've heard that CS is better. But still has tendencies that way.
I've heard that CS is better. But still has tendencies that way.
I understand this confusion, but it totally isn't how research is done. It's very much a community and social effort; it can be very lonely working all by yourself on something.
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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...
Well, parts of it at least [1].
1: http://research.microsoft.com/en-us/news/features/gonthierpr...
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.)
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.)
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.
It probably just means to say "holds true for every regular polyhedron".
Are finite simple groups specifically about polyhedra, or was that just used as an example?
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.
The monster is, for example, the symmetry group of an object in 196884 dimensions. It's not really a polyhedron.
"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.