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. 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).