finitely generated group

A finitely generated group is a group that has a finite generating set.

Every finite group is obviously finitely generated.Every finitely generated group is countable.

Any quotient (http://planetmath.org/QuotientGroup)of a finitely generated group is finitely generated.However, a finitely generated group may have subgroupsthat are not finitely generated.(For example, the free group of rank $2$ is generated by just two elements,but its commutator subgroup is not finitely generated.)Nonetheless, a subgroup of finite index in a finitely generated groupis necessarily finitely generated;a bound on the number of generators required for the subgroup is given bythe Schreier index formula (http://planetmath.org/ScheierIndexFormula).

The finitely generated groupsall of whose subgroups are also finitely generatedare precisely the groups satisfying the maximal condition.This includes all finitely generated nilpotent groups and,more generally, all polycyclic groups.

A group that is not finitely generatedis sometimes said to be infinitely generated.