text: While it is true that every quotient of a finitely generated group is finitely generated (the images of the generators in the quotient give a finite generating set), a subgroup of a finitely generated group need not be finitely generated. For example, let  be the free group in two generators,  and  (which is clearly finitely generated, since ), and let  be the subset consisting of all elements of  of the form  for some natural number .  is isomorphic to the free group in countably infinitely many generators, and so cannot be finitely generated. However, every subgroup of a finitely generated abelian group is in itself finitely generated. In fact, more can be said: the class of all finitely generated groups is closed under extensions. To see this, take a generating set for the (finitely generated) normal subgroup and quotient. Then the generators for the normal subgroup, together with preimages of the generators for the quotient, generate the group.
  Examples
  The multiplicative group of integers modulo 9, , is the group of all integers relatively prime to 9 under multiplication . Note that 7 is not a generator of , since     while 2 is, since
  On the other hand, Sn, the symmetric group of degree n, is not generated by any one element (is not cyclic) when n > 2. However, in these cases Sn can always be generated by two permutations which are written in cycle notation as (1 2) and . For example, the 6 elements of S3 can be generated from the two generators, (1 2) and (1 2 3), as shown by the right hand side of the following equations (composition is left-to-right):
  e = (1 2)(1 2)
  (1 2) = (1 2)
  (1 3) = (1 2)(1 2 3)
  (2 3) = (1 2 3)(1 2)
  (1 2 3) = (1 2 3)
  (1 3 2) = (1 2)(1 2 3)(1 2)