Non-Abelian Simple Finite Groups are Infinitely Many

Theorem
There exist infinitely many types of group which are non-abelian and finite.

Proof
We have that Alternating Group is Simple except on 4 Letters.

So for all $n \in \N$ such that $n \ne 4$, the alternating group $A_n$ is a simple group.

We also have that $A_n$ is non-abelian for all $n > 3$.

Hence the result.