For those of you who are a bit rusty: a finite group is a group that has a finite number of elements. A simple group is one that has no proper non-trivial normal subgroups (that is, only the identity and the whole group are normal subgroups).
It is a theorem that if is a finite simple group then falls into one of the following categories:
1. Cyclic (of prime order, of course)
2. Alternating (and not isomorphic to of course)
3. A member of a subclass of Lie Groups
4. One of 26 other groups that don’t fall into 1, 2 or 3.
Scientific American has a nice article about this theorem and the effort to get it written down and understood; the problem is that the proof of such a theorem is far from simple; it spans literally hundreds of research articles and would take thousands of pages to be complete. And, those who have an understanding of this result are aging and won’t be with us forever.
Here is a link to the preview of the article; if you don’t subscribe to SA it is probably in your library.