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I don't have much experience in group theory beyond knowing what a finite simple group is, as well as the other basic parts of the topic, so far as I can tell, so if the actual answer is too complex to be summarised clearly then just say and I'll understand that.

So far as I can tell the classification of finite simple groups states that if you were to find a finite simple group of order n, it would either belong to one of the four families or is one of the 26 sporadic simple groups.

Is it proven then that there are no more possible families of finite simple groups, so incredibly large that we couldn't find them with computation?

Is it proven that all of the sporadic simple groups smaller than the monster have been found, and that there cannot be any greater ones?

If the answer is yes for either one or both of those questions, then is there any way to summarise why in fairly simple terms?

If the answer is no for either then what is the mathematical community's opinion on that?

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    $\begingroup$ Yes, it would not be called a classification if it had not included proofs. Unfortunately, the complete proof is ridiculously long. $\endgroup$ – Tobias Kildetoft Aug 18 '14 at 18:49
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    $\begingroup$ Maybe this article is of interest for you. $\endgroup$ – j.p. Aug 19 '14 at 8:25
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    $\begingroup$ basically a duplicate of this question (i would close if the other question had an answer, but really, just read the article j.p. linked) $\endgroup$ – Alexander Gruber♦ Aug 20 '14 at 18:43
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To quote Wikipedia: "The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof."

Summarize in simple terms? I'd be surprised if somebody here could do that... :)

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  • $\begingroup$ Actually I think the link you have given does about the best job you can do - the timeline, for example. There are longer and more basic accounts of the story (e.g. Mark Ronan's book Symmetry and The Monster) - but as a way in to some of the mathematical ideas involved (many of which are non-trivial and have to be skated over in an elementary treatment) it seemed a good start to me. $\endgroup$ – Mark Bennet Aug 18 '14 at 19:19

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