Simple groups and the classification of finite groups

How can we describe all finite groups? Before we address this question, let’s write down a list of all the finite groups of small orders ≤ 15, up to isomorphism. We have seen almost all of these already. If G is abelian, it is easy to write down all possible G of a given order, using the Fundamental Theorem of Finite Abelian Groups: G must be isomorphic to a direct product of cyclic groups, and any isomorphism between two such direct products is a consequence of the Chinese Remainder Theorem. For example, if #(G) = n and n is a product of distinct primes then G is cyclic, and hence isomorphic to Z/nZ. The only cases where this doesn’t happen for n ≤ 15 are: • n = 4 and G ∼= Z/4Z or G ∼= (Z/2Z)× (Z/2Z). • n = 8 and G ∼= Z/8Z, G ∼= (Z/4Z)×(Z/2Z) or G ∼= (Z/2Z)×(Z/2Z)× (Z/2Z).