I've never been able to find any details on what exactly decides what the classifications ought to be for finite simple groups. We have:
- Cyclic groups
- Alternating groups
- Groups of Lie type
- Sporadic groups
But why does the classification stop there? If there are no requirements on what the categories should be, I could classify all finite simple groups by saying "they're all finite and simple" or "they're all either cyclic or non-cyclic". Or maybe the above categories could be broken down even further into sub-categories. Why was it decided that these ought to be the categories into which we classify groups?
As a subquestion: what determines whether or not a group of "sporadic"? Who's to say that the existence of sporadic groups isn't just a sign that the classification is bad, and that the categories need to be redefined so as to include a place for the sporadic groups?