Definition:Generated Subgroup/Definition 3

Definition

Let $G$ be a group.

Let $S \subset G$ be a subset.

Let $S^{-1}$ be the set of inverses of $S$.

The subgroup generated by $S$ is the set of words on the union $S \cup S^{-1}$.

Also see

• Equivalence of Definitions of Generated Subgroup

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.3$. Subgroup generated by a subset
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{K}$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms