Is finite group theory still a fruitful area of research? 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?
 A: 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 .
A: 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.
