Editing Generating set of a group (section)

==Examples==
* The [[Multiplicative_group_of_integers_modulo_n|multiplicative group of integers modulo 9]], {{math|1=U<sub>9</sub>&nbsp;=&nbsp;{{mset|1,&nbsp;2,&nbsp;4,&nbsp;5,&nbsp;7,&nbsp;8}}}}, is the group of all integers [[Coprime|relatively prime]] to&nbsp;9 under multiplication {{math|1=[[Modular arithmetic|mod]]&nbsp;9}}. Note that 7 is not a generator of {{math|U<sub>9</sub>}}, since <br /> &nbsp; <math>\{7^i \bmod{9}\ |\ i \in \mathbb{N}\} = \{7,4,1\},</math> <br />while 2 is, since <br /> &nbsp; <math>\{2^i \bmod{9}\ |\ i \in \mathbb{N}\} = \{2,4,8,7,5,1\}.</math>

* On the other hand, ''S''<sub>n</sub>, the [[symmetric group]] of degree ''n'', is not generated by any one element (is not [[Cyclic_group|cyclic]]) when ''n'' > 2. However, in these cases ''S''<sub>n</sub> can always be generated by two permutations which are written in [[Permutation#Cycle_notation|cycle notation]] as (1 2) and {{math|1=(1&nbsp;2&nbsp;3&nbsp;...&nbsp;''n'')}}. For example, the 6 elements of ''S''<sub>3</sub> 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)

* Infinite groups can also have finite generating sets. The additive group of integers has&nbsp;1 as a generating set. The element&nbsp;2 is not a generating set, as the odd numbers will be missing. The two-element subset {{math|1={{mset|3,&nbsp;5}}}} is a generating set, since {{math|1=(&minus;5)&nbsp;+&nbsp;3&nbsp;+&nbsp;3&nbsp;=&nbsp;1}} (in fact, any pair of [[Coprime integers|coprime]] numbers is, as a consequence of [[Bézout's identity]]).

* The [[dihedral group]] of an [[Polygon|n-gon]] (which has [[Order_(group_theory)|order]] {{math|1=2n}}) is generated by the set {{math|1={{mset|{{var|r}}, {{var|s}}}}}}, where {{mvar|r}} represents rotation by {{math|1=2''π''/{{var|n}}}} and {{mvar|s}} is any reflection across a line of symmetry.<ref>{{Cite book|title=Abstract algebra|last=Dummit |first=David S.|date=2004|publisher=Wiley|last2=Foote |first2=Richard M. |isbn=9780471452348|edition=3rd |oclc=248917264|page=25}}</ref>

* The [[cyclic group]] of order <math>n</math>, <math>\mathbb{Z}/n\mathbb{Z}</math>, and the <math>n</math><sup>th</sup> [[Root of unity|roots of unity]] are all generated by a single element (in fact, these groups are [[Group isomorphism|isomorphic]] to one another).<ref>{{harvnb|Dummit|Foote|2004|p=54}}</ref>

* A [[presentation of a group]] is defined as a set of generators and  a collection of relations between them, so any of the examples listed on that page contain examples of generating sets.<ref>{{harvnb|Dummit|Foote|2004|p=26}}</ref>