Finitely generated group
In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the product of finitely many elements of the finite set S and of inverses of such elements.
By definition, every finite group is finitely generated, since S can be taken to be G itself. Every infinite finitely generated group must be countable but countable groups need not be finitely generated. The additive group of rational numbers Q is an example of a countable group that is not finitely generated.
Every quotient of a finitely generated group is finitely generated. A subgroup of a finitely generated group need not be finitely generated.
A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z.
A group is locally cyclic if every finitely generated subgroup is cyclic. The additive group of the rational numbers Q is an example of a non-cyclic locally cyclic group. Every locally cyclic group is abelian. Every finitely generated locally cyclic group is cyclic.
Every abelian group can be seen as a module over the ring of integers Z, and in a finitely generated abelian group with generators x1, ..., xn, every group element x can be written as a linear combination of these generators,
...
Wikipedia