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A while back I was learning about finite groups in an algebra class. I mentioned to a friend that finite group theory might be an interesting area of research to pursue. She asked something along the lines of "Isn't finite group theory 'done?' I thought that ended with the classification theorems."

I'm not sure how to answer. Is finite group theory still an active area of research? If so, what are some of the common problems that professors and graduate students are working on right now? Are there any reading materials to get more acquainted with such things?

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I did some research on prime graphs of finite groups lately. I wouldn't say finite group theory is as active as before the classification theorems but there's still some smaller stuff going on. A lot of it is tied up with representation theory, for example monomial groups are still actively studied. –  Alexander Gruber Oct 16 '12 at 20:05
    
If you are interested, how a much improved proof of the classification of finite simple groups could look like, take a look at math.msu.edu/~meier/Preprints/CGP/cgp_abstract.html [This is a still active area, but technically quite difficult.] –  j.p. Oct 16 '12 at 20:45
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ams.org/mathscinet/search/… –  Jack Schmidt Oct 17 '12 at 13:51
    
@JackSchmidt I guess your comment means that it is still a fruitful area of research (because there are so many publications later than 2009), but your link seems to require a subscription. –  Thomas Klimpel Oct 17 '12 at 14:56
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What about the (modular) characters of the FSG? Or the McKay conjecture? Or the cohomology of p-groups? Or minimal presentations of groups? Or... There are a fantastic amount of open questions, unexplored areas, and active research projects. Plus there are areas that "spring" from FGT, like fusion systems, subgroup complexes, cohomology varieties, etc. Certainly very very fruitful. –  user641 Oct 17 '12 at 16:02

2 Answers 2

I think it's a good idea if you connect group theory with some other branch of mathematics. For example, geometric group theory is quite active in recent days,

http://en.wikipedia.org/wiki/Geometric_group_theory .

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Geometric group theory mainly studies infinite groups: every finite groups are quasi-isometric. –  Seirios Oct 17 '12 at 10:17

I have no idea about the level of activity in finite group theory. You may safely assume that the easy answers have already be found (as well as many challenging answers), and what is left is really challenging.

Note that the classification theorem was only the first part of the Hölder program. The second part is considered to be even more challenging than the first part, but we can't even proof that it is unrealistic. Some simple questions related to the second part of the Hölder program have even appeared on this site (like Classifications of finite nilpotent groups or Quaternion group as an extension, and you would probably be able to find much more on MSE and MSO). You can see from the reactions that there is still great interest in answers to these questions.

There is definitively much more to finite groups than just the Hölder program, but this should demonstrate at least that there are still many unanswered interesting questions.

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