# Is the classification of finite groups not a bit arbitrary?

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

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?

• " If there are no requirements on what the categories should be" ??? – Martin Brandenburg Oct 18 '13 at 12:07
• So what you're asking is more or less how specific one needs to be in order to justify calling a result a full classification? – Arthur Oct 18 '13 at 12:09
• @Arthur Basically, yes. Is there a rigorous definition somewhere of what a "classification" is, or did mathematicians just stop classifying when they reached a level of granularity that was useful to them? – Jack M Oct 18 '13 at 12:11
• I think there is a fundamental misunderstanding here: "non-cyclic" does not uniquely determine a group; so, classification theorem tells us list of properties and the group these properties uniquely determine; conversely, and more importanly, every group that you can think of appears in this list (of course, upto isomorphism); classification is not just putting groups under different headings; today, every group can be identified upto isomorphism by a list of properties. Conversely, in principle, one should be able to determine all groups upto isomorphism that satisfy a given property.[contd] – knsam Oct 18 '13 at 12:16
• [contd] Of course all these comments apply to finite simple groups. (So, I meant a finite simple group when I meant groups in the above comment!) – knsam Oct 18 '13 at 12:19

Say I want the finite, simple groups of order $n$. Then I just look at the cyclic $n$-group (easily constructed, ant it's simple when $n$ is prime). Then I go on to the alternating group of order $n$ (exists for $n = m!/2$ for some $m$, and simple for $m = 3$ or $m \geq 5$). And so on for Lie type groups and sporadic groups. I can give you all simple groups of a given order, because they have all been classified.