What is simple group?

A group $G \neq \{e\}$ is simple if its only normal subgroups are $\{e\}$ and $G$. A nonabelian simple group is not solvable: $D(G) \trianglelefteq G$ is not $\{e\}$ (else $G$ abelian), so $D(G) = G$ and the derived series is constant. The abelian simple groups are exactly the $\Z/p\Z$, $p$ prime (an abelian group is simple iff it has no proper nontrivial subgroup, iff it is cyclic of prime order by Lagrange).