View source for Generating set of a group

{{Short description|Abstract algebra concept}}
[[File:One5Root.svg|thumb|The 5th roots of unity in the complex plane form a [[Group (mathematics)|group]] under multiplication. Each non-identity element generates the group.]]
In [[Abstract algebra|abstract algebra]], a '''generating set of a group''' is a [[Subset|subset]] of the group set such that every element of the [[Group (mathematics)|group]] can be expressed as a combination (under the group operation) of finitely many elements of the subset and their [[Inverse element|inverses]].

In other words, if <math>S</math> is a subset of a group <math>G</math>, then <math>\langle S\rangle</math>, the ''subgroup generated by <math>S</math>'', is the smallest [[Subgroup|subgroup]] of <math>G</math> containing every element of <math>S</math>, which is equal to the intersection over all subgroups containing the elements of <math>S</math>; equivalently, <math>\langle S\rangle</math> is the subgroup of all elements of <math>G</math> that can be expressed as the finite product of elements in <math>S</math> and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.)

If <math>G=\langle S\rangle</math>, then we say that <math>S</math> ''generates'' <math>G</math>, and the elements in <math>S</math> are called ''generators'' or ''group generators''. If <math>S</math> is the empty set, then <math>\langle S\rangle</math> is the [[Trivial group|trivial group]] <math>\{e\}</math>, since we consider the empty product to be the identity.

When there is only a single element <math>x</math> in <math>S</math>, <math>\langle S\rangle</math> is usually written as <math>\langle x\rangle</math>. In this case, <math>\langle x\rangle</math> is the ''cyclic subgroup'' of the powers of <math>x</math>, a [[Cyclic group|cyclic group]], and we say this group is generated by <math>x</math>. Equivalent to saying an element <math>x</math> generates a group is saying that <math>\langle x\rangle</math> equals the entire group <math>G</math>. For [[Finite group|finite group]]s, it is also equivalent to saying that <math>x</math> has [[Order (group theory)|order]] <math>|G|</math>.

A group may need an infinite number of generators. For example the additive group of [[Rational number|rational number]]s <math>\Q</math> 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&nbsp;− see [[Frattini subgroup]] below.

If <math>G</math> is a [[Topological group|topological group]] then a subset <math>S</math> of <math>G</math> is called a set of ''topological generators'' if <math>\langle S\rangle</math> is [[Dense set|dense]] in <math>G</math>, i.e. the [[Closure (topology)|closure]] of <math>\langle S\rangle</math> is the whole group <math>G</math>.

==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|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|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=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|subgroup]] of a finitely generated group need not be finitely generated. For example, let <math>G</math> be the [[Free group|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|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|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|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 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 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|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|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|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>

==Free group==
{{Main|Free group}}
The most general group generated by a set <math>S</math> is the group [[Free group|freely generated]] by <math>S</math>. Every group generated by <math>S</math> is isomorphic to a [[Quotient group|quotient]] of this group, a feature which is utilized in the expression of a group's presentation.

==Frattini subgroup==
An interesting companion topic is that of ''non-generators''. An element <math>x</math> of the group <math>G</math> is a non-generator if every set <math>S</math> containing <math>x</math> that generates <math>G</math>, still generates <math>G</math> when <math>x</math> is removed from <math>S</math>. In the integers with addition, the only non-generator is 0. The set of all non-generators forms a subgroup of <math>G</math>, the [[Frattini subgroup]].

==Semigroups and monoids==
If <math>G</math> is a [[Semigroup|semigroup]] or a [[Monoid|monoid]], one can still use the notion of a generating set <math>S</math> of <math>G</math>. <math>S</math> is a semigroup/monoid generating set of <math>G</math> if <math>G</math> is the smallest semigroup/monoid containing <math>S</math>. 

The definitions of generating set of a group using finite sums, given above, must be slightly modified when one deals with semigroups or monoids. Indeed, this definition should not use the notion of inverse operation anymore. The set <math>S</math> is said to be a semigroup generating set of <math>G</math> if each element of <math>G</math> is a finite sum of elements of <math>S</math>. Similarly, a set <math>S</math> is said to be a monoid generating set of <math>G</math> if each non-zero element of <math>G</math> is a finite sum of elements of <math>S</math>.

For example, {1} is a monoid generator of the set of [[Natural number|natural number]]s <math>\N</math>. The set {1} is also a semigroup generator of the positive natural numbers <math>\N_{>0}</math>. However, the integer 0 can not be expressed as a (non-empty) sum of 1s, thus {1} is not a semigroup generator of the natural numbers. 

Similarly, while {1} is a group generator of the set of [[Integer|integer]]s <math>\mathbb Z</math>, {1} is not a monoid generator of the set of integers. Indeed, the integer −1 cannot be expressed as a finite sum of 1s.

==See also==
* Generating set for related meanings in other structures
* Presentation of a group
* [[Primitive element (finite field)]]
* [[Cayley graph]]

==Notes==
<references/>

==References==
* {{Lang Algebra|edition=3r}}
*{{cite book |last1=Coxeter |first1=H.S.M.  |last2=Moser |first2=W.O.J.  | title=Generators and Relations for Discrete Groups | publisher=Springer| year=1980 | isbn=0-387-09212-9}}

== External links ==
*{{mathworld |urlname=GroupGenerators |title=Group generators}}

[[Category:Group theory]]

{{Sourceattribution|Generating set of a group}}