Generating set of a group

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.

In other words, if S is a subset of a group G, then S, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S; equivalently, S is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses. (Notice: here inverses are only needed if the group is infinite. For a finite group, the inverse can be expressed as a power of the element itself.)

If G = S, then we say that S generates G, and the elements in S are called generators or group generators. If S is the empty set, then S is the trivial group {e}, since we consider the empty product to be the identity.

When there is only a single element x in S, S is usually written as x. In this case, x is the cyclic subgroup of the powers of x, a cyclic group, and we say this group is generated by x. Equivalent to saying an element x generates a group is saying that x equals the entire group G. For finite groups, it is also equivalent to saying that x has order |G|.

If G is a topological group then a subset S of G is called a set of topological generators if S is dense in G i.e. the closure of S is the whole group G.

Finitely generated group

If S is finite, then a group G = S is called finitely generated. The structure of finitely 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 S, then each group element may be expressed as a word from the alphabet S of length less than or equal to the order of the group.

Every finite group is finitely generated since G = G. The integers under addition are an example of an infinite group which is finitely generated by both 1 and −1, but the group of rationals under addition cannot be finitely generated. No uncountable group can be finitely generated. For example, the group of real numbers under addition, (R, +).

Different subsets of the same group can be generating subsets; for example, if p and q are integers with gcd(p, q) = 1, then {p, q} also generates the group of integers under addition (by Bézout's identity).

While it is true that every quotient of a finitely generated group is finitely generated (simply take the images of the generators in the quotient), a subgroup of a finitely generated group need not be finitely generated. For example, let G be the free group in two generators, x and y (which is clearly finitely generated, since G = {x,y}), and let S be the subset consisting of all elements of G of the form ynxyn, for n a natural number. Since S is clearly isomorphic to the free group in countably infinite number of generators, it 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.

Free group

The most general group generated by a set S is the group freely generated by S. Every group generated by S is isomorphic to a 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 x of the group G is a non-generator if every set S containing x that generates G, still generates G when x is removed from S. In the integers with addition, the only non-generator is 0. The set of all non-generators forms a subgroup of G, the Frattini subgroup.

Examples

The group of units U(Z9) is the group of all integers relatively prime to 9 under multiplication mod 9 (U9 = {1, 2, 4, 5, 7, 8}). All arithmetic here is done modulo 9. Seven is not a generator of U(Z9), since while 2 is, since: On the other hand, for n > 2 the symmetric group of degree n is not cyclic, so it is not generated by any one element. However, it is generated by the two permutations (1 2) and (1 2 3 ... n). For example, for S3 we have:

e = (1 2)(1 2)
(1 2) = (1 2)
(2 3) = (1 2)(1 2 3)
(1 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 of coprime numbers is, as a consequence of Bézout's identity).

The dihedral group of order n is generated by the set {r, s}, where r represents rotation by π/n and s is any reflection about a line of symmetry.

The cyclic group of order n, , and the nth roots of unity are all generated by a single element (in fact, these groups are isomorphic to one another).

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.

Semigroups and monoids

If G is a semigroup or a monoid, one can still use the notion of a generating set S of G. S is a semigroup/monoid generating set of G if G is the least semigroup/monoid containing S.

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

For example {1} is a monoid generator of the set of non-negative natural numbers . The set {1} is also a semigroup generator of the positive natural numbers . However, the integer 0 can not be expressed as a (non-empty) sum of 1's, thus {1} is not a semigroup generator of the non-negative natural numbers.

Similarly, while {1} is a group generator of the set of relative integers , {1} is not a monoid generator of the set of relative integers. Indeed, the integer -1 can not be expressed as a finite sum of 1's.