How did we classify the finite simple groups when we haven't classified the primes?

Question

Why was the classification of the finite non-abelian simple groups "easier" (!!) than the classification of the finite abelian simple groups [the prime numbers], which still doesn't exist?

(Does it? There are infeasible algorithms for generating a complete irredundant list of prime numbers, but then it's not that hard to come up with an infeasible algorithm to "classify" all finite simple groups the same way.. construct each group and test for simplicity, only different is you need to then check isomorphism with other groups of same order to eliminate duplicates, but there's an infeasible algorithm for that too..)

And actually, what is the mathematical meaning of "classification"? Since the "classification" of non-abelian finite simple groups doesn't actually give a complete irredundant list (like the way, say, the classification of closed surfaces does), since 13 of the families of Lie type are indexed by prime numbers.

Answer 1

The finite abelian simple groups are exactly $\mathbb Z / p\mathbb Z$ for $p$ a prime. If we want to classify finite simple groups then once we've put all of these groups in one class we are done with the abelian finite simple groups and we move on. We don't need to understand the primes to know that these are all the abelian finite simple groups.

Answer 2

Perhaps someone has already mentioned this in the comments, but it should be pointed out that even the statement of the classification of finite nonabelian simple groups is heavily dependent on the classification of primes, because the groups of Lie type, which constitute the vast majority of the finite simple groups, all have an arbitrary prime power as a parameter.