@Jack don't worry, I can provide you with a few links as soon as you reach your 10k :) however, they're not as awesome as you might think... It's been quite calm recently after the fury of some webhost a while ago
I'll upvote your transitive automorphism groups implies abelian answer tomorrow, then :) rep-capping is quite annoying...
Wait, you've got 4 accepts, so you'll have three more counting votes.
I always saw that as sort of Ockham's razor and a test if the "pure" group theoretical tools are sharp enough. But usually those character-free proofs are so much more complicated. Is it just to illustrate that representation theory is so powerful or is there another reason?
I think it is partially because character theory proofs often use congruences and inequalities and number shuffling of all sorts (sort of like the congruence part of Sylow theorem trickery). At the end of most of these proofs, we just get a contradiction. We don't learn anything about the group, as all the intermediate steps are about some meaningless numbers.
Ah, that makes a lot of sense. That never occurred to me. So the hope is to get deeper insight in the inner workings of why the facts are true by looking for something more constructive. Great! Now I know what to look for when comparing. Thanks!
However, the character proofs of Burnside's theorem and Frobenius's theorem are both very short and beautiful. So nobody really minded them. But then Goldschmidt and (I think) more importantly Bender came up with methods of handling Burnside's theorem that actually described deep features of the groups.
So now people are hoping for more insight by trying other problems. Frobenius is high profile because the original proof is so good, the statement is pretty simple, and no one has managed to solve it. Maybe like Fermat's Last Theorem: its so simple, surely I can do it.
On the flip side though: lots of character theory proofs are about linear groups, and tell us about linear groups, and give us a deeper understanding of groups. I think everyone likes those. Its just the crazy numerical finishes of proofs like the the odd order theorem that make people nervous. Feit and Thompson actually used calculus in their proof (scary!).
That's excellent and provides me with some food for thought, thanks a lot! (I don't mind calculus, of course :)) But I'm a very non-constructive guy...
Could you point me to a good exposition of these ideas of Goldschmidt and Bender (for someone not so much acquainted with finite groups?)
I like Isaacs's Finite Group Theory book (page 216, I was just checking if it was a good rec). If you don't mind original sources, look at Bender's (partial) characterization of simple groups with Klein-4 Sylow 2-subgroups (i can get you the paper reference). It is frighteningly simple compared to gorenstein-walter.
@tb: ams.org/mathscinet-getitem?mr=335634 is an easy Bender paper and ams.org/mathscinet-getitem?mr=1879251 is a modern (2001) adaptation that uses more or less the same methods for a strengthened result. i mention because bender has similar sounding papers that are easier than Gorenstein–Walter but not easy
I actually find it a little hard to imagine life (in the finite group theory sense) before Bender. What did people even think groups looked like before they had his version of the Fitting subgroup? You had no internal way of representing the group at all.
@Jack: Thanks a lot, again! I think that's more than enough for now. I'll make sure to try and understand what you're telling me in the last paragraph and I'll keep it in mind. Since I don't know yet what Bender did, I don't think I can appreciate that statement fully.
There's a way you can find out how many times you've capped: If you're logged in, go to math.stackexchange.com/reputation. It will tell you at the bottom how many days you've hit 200 rep.
You can also query data.stackexchange.com/mathematics/queries to find out how much total rep you've lost because of rep caps. I don't remember what the query is named, but it's in there.
@robjohn: Yes, although when I've tried I've usually messed up something in the syntax and so my query failed to parse. I haven't taken the time to learn the language well enough, and so I've just used ones others have created.
heh :) yes, it's almost one here. I was working the whole day and found an obscure reference containing many of the things that I've been working on over the past months.
No, I'm working on a rather bigger thing and it was part of work-in-progress. While looking up something completely unrelated, I found a paper from a few years ago containing much of the things I've been doing until today.
@robjohn: just to finish the former thing up, there's some disappointment but the neat thing is that I can answer a few questions they raised, so there definitely remains something from that work. On the other hand they answer stuff that I haven't thought about yet, so in the end there will be a positive outcome. It's just momentary disappointment for a bit of wasted time.
One problem I've always hit is that stuff that seems new to you and your colleagues turns out to have been discussed (and extensively at that!) in a paper in a journal you and your colleagues wouldn't be usually peeking at...
...which gives you the feeling the author of that paper would consider you as unwashed heathens...
Since I am only an amateur, I consider anything new I find to be exciting. Most of the things I prove I can find elsewhere. The last new thing I discovered was a couple of years ago.
Yes, exactly. It happened several times already that I had to referee things in which some of the main theorems were sort of folklore lemmas for people in my surroundings but were never written up properly or were pushed so much further that the main idea was blurred.
(I was responding to JM's last point from the opposite perspective)
Since I'm being talked about, I suppose it's only polite to mention that I'm here. :) And I agree, robjohn, about us approaching combinatorial problems in a similar way.
@robjohn: It's edits to a paper. Believe it or not, it's based off of this old math.SE post. It's a survey paper composed of the answers to that question.
@J.M.: I agree. I may be a math professor, but I've learned as much in the past year from math.SE as I have in any year since I left grad school, I think.
@J.M.: I proved some sharp results with pseudodifferential operators using difference operators rather than differential operators. I know these were new because my advisor, who knows everything about pseudo DEs said they were.
While I'm on the subject of benefits from math.SE... It has helped keep me sharp in the areas on the site I'm more active in, and I'm pretty sure it's improved my ability to communicate mathematics in writing. I think I saw Arturo say something similar to my last point once, too.
@robjohn: It's heavier going, but you might like the other paper I mention in that same blog post. It gives a partial answer to one of the research problems in Concrete Mathematics. (I'm guessing you're also a fan of CMath.)
@J.M.: I got a second copy because my first one was falling apart. Now I have one at home and at work, so can use it to answer math.SE questions no matter where I am. :)
