Definition:Generated Subgroup/Definition 1

Definition

Let $G$ be a group.

Let $S \subset G$ be a subset.

The subgroup generated by $S$ is the smallest subgroup containing $S$.

Also see

• Equivalence of Definitions of Generated Subgroup

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $5$: Subgroups: Exercise $11$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{K}$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 37.5$ Some important general examples of subgroups
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $4$: Subgroups: Definition $4.7$