Definition:Generated Subgroup
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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