Finitely generated group

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


 * 1) It has a finite generating set.
 * 2) Every generating set of the group has a subset that is finite and is also a generating set.
 * 3) The group has at least one minimal generating set and every minimal generating set of the group is finite.
 * 4) The minimum size of generating set of the group is finite.
 * 5) The group is a join of finitely many cyclic subgroups.

Opposite properties

 * Locally finite group is a group where every finitely generated subgroup is finite. A group is locally finite and finitely generated if and only if it is finite.

Effect of property operators
A slender group, or Noetherian group, is a group such that all its subgroups are finitely generated.