text: A group may need an infinite number of generators. For example the additive group of rational numbers  is not finitely generated. It is generated by the inverses of all the integers, but any finite number of these generators can be removed from the generating set without it ceasing to be a generating set. In a case like this, all the elements in a generating set are nevertheless "non-generating elements", as are in fact all the elements of the whole group − see Frattini subgroup below.
  If  is a topological group then a subset  of  is called a set of topological generators if  is dense in , i.e. the closure of  is the whole group .
  Finitely generated group
  If  is finite, then a group  is called finitely generated. The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general. It has been proven that if a finite group is generated by a subset , then each group element may be expressed as a word from the alphabet  of length less than or equal to the order of the group.
  Every finite group is finitely generated since . The integers under addition are an example of an infinite group which is finitely generated by both 1 and −1, but the group of rationals under addition cannot be finitely generated. No uncountable group can be finitely generated. For example, the group of real numbers under addition, .
  Different subsets of the same group can be generating subsets. For example, if  and  are integers with , then  also generates the group of integers under addition by Bézout's identity.