Inabstract algebra, agenerating set of a group is asubset of the group set such that every element of thegroup can be expressed as a combination (under the group operation) of finitely many elements of the subset and theirinverses.

In other words, if${\displaystyle S}$ is a subset of a group${\displaystyle G}$, then${\displaystyle ⟨S⟩}$, thesubgroup generated by${\displaystyle S}$, is the smallestsubgroup of${\displaystyle G}$ containing every element of${\displaystyle S}$, which is equal to the intersection over all subgroups containing the elements of${\displaystyle S}$; equivalently,${\displaystyle ⟨S⟩}$ is the subgroup of all elements of${\displaystyle G}$ that can be expressed as the finite product of elements in${\displaystyle S}$ 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${\displaystyle G=⟨S⟩}$, then we say that${\displaystyle S}$generates${\displaystyle G}$, and the elements in${\displaystyle S}$ are calledgenerators orgroup generators. If${\displaystyle S}$ is the empty set, then${\displaystyle ⟨S⟩}$ is thetrivial group${\displaystyle \{e\}}$, since we consider theempty product to be the identity.

When there is only a single element${\displaystyle x}$ in${\displaystyle S}$,${\displaystyle ⟨S⟩}$ is usually written as${\displaystyle ⟨x⟩}$. In this case,${\displaystyle ⟨x⟩}$ is thecyclic subgroup of the powers of${\displaystyle x}$, acyclic group, and we say this group is generated by${\displaystyle x}$. Equivalent to saying an element${\displaystyle x}$ generates a group is saying that${\displaystyle ⟨x⟩}$ equals the entire group${\displaystyle G}$. Forfinite groups, it is also equivalent to saying that${\displaystyle x}$ hasorder${\displaystyle |G|}$.

A group may need an infinite number of generators. For example the additive group ofrational numbers${\displaystyle \mathbb{Q}}$ 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 − seeFrattini subgroup below.

If${\displaystyle G}$ is atopological group then a subset${\displaystyle S}$ of${\displaystyle G}$ is called a set oftopological generators if${\displaystyle ⟨S⟩}$ isdense in${\displaystyle G}$, i.e. theclosure of${\displaystyle ⟨S⟩}$ is the whole group${\displaystyle G}$.

If${\displaystyle S}$ is finite, then a group${\displaystyle G=⟨S⟩}$ is calledfinitely generated. The structure offinitely 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${\displaystyle S}$, then each group element may be expressed as a word from the alphabet${\displaystyle S}$ of length less than or equal to the order of the group.

Every finite group is finitely generated since${\displaystyle ⟨G⟩=G}$. Theintegers under addition are an example of aninfinite group which is finitely generated by both 1 and −1, but the group ofrationals under addition cannot be finitely generated. Nouncountable group can be finitely generated. For example, the group of real numbers under addition,${\displaystyle (\mathbb{R},+)}$.

Different subsets of the same group can be generating subsets. For example, if${\displaystyle p}$ and${\displaystyle q}$ are integers withgcd(p, q) = 1, then${\displaystyle \{p,q\}}$ also generates the group of integers under addition byBézout's identity.

While it is true that everyquotient of afinitely generated group is finitely generated (the images of the generators in the quotient give a finite generating set), asubgroup of a finitely generated group need not be finitely generated. For example, let${\displaystyle G}$ be thefree group in two generators,${\displaystyle x}$ and${\displaystyle y}$ (which is clearly finitely generated, since${\displaystyle G=⟨\{x,y\}⟩}$), and let${\displaystyle S}$ be the subset consisting of all elements of${\displaystyle G}$ of the form${\displaystyle y^{n}x{y}^{-n}}$ for somenatural number${\displaystyle n}$.${\displaystyle ⟨S⟩}$ isisomorphic to the free group in countably infinitely many generators, and so cannot be finitely generated. However, every subgroup of a finitely generatedabelian group is in itself finitely generated. In fact, more can be said: the class of all finitely generated groups is closed underextensions. 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.

• Themultiplicative group of integers modulo 9,U₉ = {1, 2, 4, 5, 7, 8}, is the group of all integersrelatively prime to 9 under multiplicationmod 9. Note that 7 is not a generator ofU₉, since
   ${\displaystyle \{7^{i}mod9|i\in \mathbb{N}\}=\{7,4,1\},}$
  while 2 is, since
   ${\displaystyle \{2^{i}mod9|i\in \mathbb{N}\}=\{2,4,8,7,5,1\}.}$

• On the other hand,S_n, thesymmetric group of degreen, is not generated by any one element (is notcyclic) whenn > 2. However, in these casesS_n can always be generated by two permutations which are written incycle notation as (1 2) and(1 2 3 ... n). For example, the 6 elements ofS₃ 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 1 as a generating set. The element 2 is not a generating set, as the odd numbers will be missing. The two-element subset{3, 5} is a generating set, since(−5) + 3 + 3 = 1 (in fact, any pair ofcoprime numbers is, as a consequence ofBézout's identity).

• Thedihedral group of ann-gon (which hasorder2n) is generated by the set{r,s}, wherer represents rotation by2π/n ands is any reflection across a line of symmetry.^[1]

• Thecyclic group of order${\displaystyle n}$,${\displaystyle \mathbb{Z}/n\mathbb{Z}}$, and the${\displaystyle n}$^throots of unity are all generated by a single element (in fact, these groups areisomorphic to one another).^[2]

• Apresentation 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.^[3]

The most general group generated by a set${\displaystyle S}$ is the groupfreely generated by${\displaystyle S}$. Every group generated by${\displaystyle S}$ isisomorphic to aquotient of this group, a feature which is utilized in the expression of a group'spresentation.

An interesting companion topic is that ofnon-generators. An element${\displaystyle x}$ of the group${\displaystyle G}$ is a non-generator if every set${\displaystyle S}$ containing${\displaystyle x}$ that generates${\displaystyle G}$, still generates${\displaystyle G}$ when${\displaystyle x}$ is removed from${\displaystyle S}$. In the integers with addition, the only non-generator is 0. The set of all non-generators forms a subgroup of${\displaystyle G}$, theFrattini subgroup.

If${\displaystyle G}$ is asemigroup or amonoid, one can still use the notion of a generating set${\displaystyle S}$ of${\displaystyle G}$.${\displaystyle S}$ is a semigroup/monoid generating set of${\displaystyle G}$ if${\displaystyle G}$ is the smallest semigroup/monoid containing${\displaystyle S}$.

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${\displaystyle S}$ is said to be a semigroup generating set of${\displaystyle G}$ if each element of${\displaystyle G}$ is a finite sum of elements of${\displaystyle S}$. Similarly, a set${\displaystyle S}$ is said to be a monoid generating set of${\displaystyle G}$ if each non-zero element of${\displaystyle G}$ is a finite sum of elements of${\displaystyle S}$.

For example, {1} is a monoid generator of the set ofnatural numbers${\displaystyle \mathbb{N}}$. The set {1} is also a semigroup generator of the positive natural numbers${\displaystyle {\mathbb{N}}_{>0}}$. 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 ofintegers${\displaystyle \mathbb{Z}}$, {1} is not a monoid generator of the set of integers. Indeed, the integer −1 cannot be expressed as a finite sum of 1s.

• Generating set for related meanings in other structures
• Presentation of a group
• Primitive element (finite field)
• Cayley graph

• Lang, Serge (2002),Algebra,Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag,ISBN 978-0-387-95385-4,MR 1878556,Zbl 0984.00001
• Coxeter, H.S.M.; Moser, W.O.J. (1980).Generators and Relations for Discrete Groups. Springer.ISBN 0-387-09212-9.

• Weisstein, Eric W."Group generators".MathWorld.

• Group theory

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