Are all non-abelian groups not simple?

Question

We know that for all $n\geq 3$ the group $S_n$ is non-abelian. We also know that for all $n\geq 3,$ the group $S_n$ is not simple, because $A_n$ is a normal subgroup of $S_n$ which is not trivial.

Can we conclude that all non-abelian groups are not simple?

Answer 1

No, we cannot. For instance, $A_n$ is simple and non-abelian for each $n\geqslant5$.