Verifying finite simple groups 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. 

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. 
