Definition:Generated Subgroup/Definition 2

Definition

Let $G$ be a group.

Let $S \subset G$ be a subset.

The subgroup generated by $S$ is the intersection of all subgroups of $G$ containing $S$.

Also see

• Intersection of Subgroups is Subgroup
• Equivalence of Definitions of Generated Subgroup

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $12$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Definition $4.7$