Subgroup generated by a subset

Symbol-free definition
Given a group, and a subset of the group, the subgroup generated by that subset is defined in the following equivalent ways:


 * It is the intersection of all subgroups containing that subset
 * It is a subgroup containing that subset, such that the subset is a generating set for the subgroup

Definition with symbols
Given a group $$G$$ and a subset $$S$$ of $$G$$, the subgroup generated by $$S$$ is defined in the following equivalent ways:


 * It is the intersection of all subgroups of $$G$$ containing $$S$$
 * It is a subgroup $$H \le G$$ such that $$S \subset H$$ and $$S$$ is a generating set for $$H$$

(Recall the fact that an intersection of subgroups is always a subgroup).

The subgroup generated by a subset $$S$$ is denoted $$\langle S \rangle$$.

Extreme examples

 * If a subset $$S$$ of a group $$G$$ is a subgroup, then $$S$$ equals the subgroup generated by $$S$$.
 * The subgroup generated by the empty subset is the trivial subgroup: it comprises only the identity element.

Other generic examples

 * The subgroup generated by a single element is the set of all its powers. In other words, the subgroup generated by an element $$g$$ is the set of all elements expressible as $$g^n, n \in \mathbb{Z}$$. This is also termed the cyclic subgroup generated by $$g$$.

Examples in Abelian groups

 * In the group of integers under addition, the subgroup generated by the integers $$4$$ and $$6$$ is the subgroup of even integers.
 * In the group of rational numbers under addition, the subgroup generated by the rational number $$1/2$$ is the group of integers and half-integers under addition.