Finitely generated simple group

Definition
A finitely generated simple group is a nontrivial group that is both a finitely generated group (i.e., it has a finite generating set) and a simple group] (i.e., it has no proper nontrivial [[normal subgroup).

This includes the groups of prime order (which are the only simple abelian groups), the finite simple non-abelian groups, and the infinite finitely generated simple groups, which must be abelian. There are infinitely many isomorphism classes of groups of each type.