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?

  • " If there are no requirements on what the categories should be" ??? – Martin Brandenburg Oct 18 '13 at 12:07
  • 1
    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
up vote 2 down vote accepted

There might not be a rigorous answer, but I believe we can say we have achieved a full classification of a type of object when we can in some way construct every single one of them and put them in front of us.

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.

As for the subquestion on the sporadic groups, after it had been found that the cyclic, alternating and Lie type groups (of appropriate order) are simple, the sporadic groups are the 26 simple, finite groups that do not fit in. Well, we know what groups they are, and it has been proven that there are only 26 of them. Without knowing the history of classification of simple groups, I suspect that that was the last result to be proven. The sporadic groups can be further divided into sub-classifications, but many of the "classes" you'd end up with only have one group in them, so it's not very useful. I understand that you might be frustrated by the fact that there's no "clean" or elegant classification, but that's how mathematics is some times.

Good question. But the classification of the simple groups just gave us these families. It is not the other way around! We cannot determine ahead what families we should have. The classification of finite simple groups is a hall mark in group theory and took years of effort and brilliant people that finally ended up with the mentioned categories. And yet these families are studied today. There are mysteries still not resolved (like Monstrous Moonshine). Still a lot of research to be done!

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