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.

$\blacksquare$

Sources

• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.11$