In abstract algebra, a generating set of a group G is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. Whilst this may appear to produce excessively long strings of terms, in fact, these strings often can be reduced. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... A is a subset of B If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X...

More generally, 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; 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. In mathematics, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation...

If G = <S>, then we say S generates G; and the elements in S are called 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> generates the cyclic subgroup of the powers of x. In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ...

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. In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...

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. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, an uncountable set is a set which is not countable. ...

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. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (GCF) or highest common factor (hcf) of two integers which are not both zero is the largest integer that divides both numbers. ...

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 factor group of this group; a feature which is utilized in the expression of a group's presentation. The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematics, one method of defining a group is by a presentation. ...

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. In mathematics, the Frattini subgroup Φ(G) of a group G is the intersection of all maximal subgroups of G. (If G has no maximal subgroups, then Φ(G) is defined to be G itself. ...

See also: Presentation of a group In mathematics, one method of defining a group is by a presentation. ...