Classification of finite simple groups

The Classification of finite simple groups is a mega-theorem which states that every finite simple group belongs to one of eighteen infinite families of simple groups, or to one of 26 sporadic simple groups.

The eighteen families

Here are the families, up to isomorphism:

1. The cyclic groups of prime order: These are the only simple Abelian groups. The set of these groups is in one-to-one correspondence with the set of prime numbers. Since there are infinitely many primes, there are infinitely many such groups. For full proof, refer: No proper nontrivial subgroup implies cyclic of prime order, prime order implies no proper nontrivial subgroup
2. The alternating groups of degree at least $5$. The alternating group of degree $n$, denoted $A_n$, is defined as the subgroup of the symmetric group on $n$ letters comprising the even permutations. The proof of their simplicity is inductive, using as base case the fact that $A_5$ is simple. For full proof, refer: A5 is simple, alternating groups are simple

The twenty-six sporadic simple groups

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