Finitely presented group

Definition
A group is said to be finitely presented or finitely presentable if it satisfies the following equivalent conditions:


 * 1) It possesses a presentation with finitely many generators, and finitely many relations.
 * 2) It is finitely generated and, for any finite generating set, it has a presentation with that generating set and finitely many relations.
 * 3) It is finitely generated and, for any presentation with a finite number of generators, we can throw away all but finitely many relations to get a presentation with finitely many generators and finitely many relations.