Definition:Generated Subgroup

Definition

Let $G$ be a group.

Let $S \subset G$ be a subset.

Definition 1

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

Definition 2

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

Definition 3

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
• Definition:Generator of Subgroup
• Definition:Generated Normal Subgroup

Sources

• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation