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The classification of finite simple groups required thousands of pages in journals. The end result is that a finite group is simple if and only if it is on a list of 26 sporadic groups and several families of groups.

Usually in classification theorems proving that the items on the list do what they're supposed to do is far simpler than proving that the list is complete. Is that the case here? How hard would it be for someone who only knows basic group theory to verify that the groups on the list really are finite simple groups?

Update: To break the question up a bit, which parts of the verification would be easiest, hardest, tedious but elementary, etc.? For example, the prime cyclic groups are simple and trivial to verify.

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see –  Alexander Gruber Sep 19 '13 at 5:09
To me it seems that way more that just basic group theory is needed even to construct all the groups on the list. –  spin Sep 19 '13 at 7:13
I do not understand why this is considered "too broad" by those voting to close. It think this is a very good, focused question. –  user1729 Sep 19 '13 at 9:13
@user1729 I consider "how hard would it be to verify the 26 sporadic groups are simple" to give rise to either very long answers, or opinionated ones. –  Lord_Farin Sep 19 '13 at 9:45
@user1729 I see where you're coming from now. On a closer look, I can see the potential added value that this question has, particularly when we can form an amalgamate of partial answers. Close vote retracted. –  Lord_Farin Sep 19 '13 at 10:28

2 Answers 2

up vote 8 down vote accepted

Probably the best source for this would be the (graduate level) textbook The Finite Simple Groups by R.A. Wilson. It is under 300 pages and covers all of the finite simple groups.

It proves simplicity of all of them.

It proves existence and uniqueness of nearly all of them.

It describes interesting structure of most of them.

I have found its explanations to be fairly simple and not to require a lot of background.

If you are only interested in some of the finite simple groups (the alternating, the classical, the chevalley, the sporadics) then there is usually a better set of books (different sets of books for each type), but if you are interested in all of them and want any hope of finishing in a timely manner, then this is the book for you!

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It is difficult. For example, if you see the proof that $PSL(2,q)$, where $q>3$ or $PSL(n,q)$, where $n>2$, (see Theorem 12.10 and Theorem 12.13 of H.E Rose, A course on finite group), is a finite simple group, is relatively difficult. Also if you see the proof that the five mathieu group are simple(see B.Huppert, N.Blackburn, Finite group III), you will see that it is difficult. As far as i know, there is no single text book that give all the proof. Also the proof differ from one class of group to the other.

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I do not think these proofs are difficult. The PSL proofs require some basic computation with transvections and some general theorems on simple groups. The Mathieu groups are treated very nicely in the last section of Passman's Permutation Groups (now reprinted by Dover). –  user641 Sep 19 '13 at 17:00
@SteveD: You are right on PSL of course, but imagine proving simplicity of E8 over a finite field with only basic knowledge of group theory! –  studiosus Sep 19 '13 at 20:08

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