Lang.org

Let G be a group and S a subset of G. We shall say that S generates G, or that S is a set of generators for G, if every element of G can be expressed as a product of elements of S or inverses of elements of S, i.e. as a product x_1 \cdots x_n, where each x_i or x_i^-1 is in S. It is clear that the set of all such products is a subgroup of G, and is the smallest subgroup of G containing S. Thus S generates G if and only if the smallest subgroup of G containing S is G itself. If G is generated by S, then we write G = \langle S \rangle