Subgroup generated by a subset

Definition

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 $H\leq G$ 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).