Editing Generating set of a group (section)

==Finitely generated group==
{{main|Finitely generated group}}
If <math>S</math> is finite, then a group <math>G=\langle S\rangle</math> is called ''finitely generated''. The structure of [[finitely generated abelian group]]s 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 <math>S</math>, then each group element may be expressed as a word from the alphabet <math>S</math> of length less than or equal to the order of the group.

Every finite group is finitely generated since <math>\langle G\rangle =G</math>. The [[integer]]s under addition are an example of an [[infinite group]] which is finitely generated by both 1 and −1, but the group of [[rational number|rationals]] under addition cannot be finitely generated. No [[uncountable]] group can be finitely generated. For example, the group of real numbers under addition, <math>(\R,+)</math>.

Different subsets of the same group can be generating subsets. For example, if <math>p</math> and <math>q</math> are integers with {{math|1=[[greatest common divisor|gcd]](''p'',&nbsp;''q'')&nbsp;=&nbsp;1}}, then <math>\{p,q\}</math> also generates the group of integers under addition by [[Bézout's identity]].

While it is true that every [[quotient group|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 <math>G</math> be the [[free group]] in two generators, <math>x</math> and <math>y</math> (which is clearly finitely generated, since <math>G=\langle \{x,y\}\rangle</math>), and let <math>S</math> be the subset consisting of all elements of <math>G</math> of the form <math>y^nxy^{-n}</math> for some [[natural number]] <math>n</math>. <math>\langle S\rangle</math> is [[Isomorphism|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 [[group extension|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.