Definition:Generated Normal Subgroup
(Redirected from Definition:Normal Closure of Subset of Group)
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Definition
Let $G$ be a group.
Let $S \subseteq G$ be a subset.
Definition 1
The normal subgroup generated by $S$, denoted $\gen {S^G}$, is the intersection of all normal subgroups of $G$ containing $S$.
Definition 2
The normal subgroup generated by $S$, denoted $\gen {S^G}$, is the smallest normal subgroup of $G$ containing $S$:
- $\gen {S^G} = \gen {x S x^{-1}: x \in G}$
Also known as
The generated normal subgroup is also known as the conjugate closure or normal closure.
Also see
- Equivalence of Definitions of Generated Normal Subgroup
- Definition:Generated Subgroup
- Definition:Contranormal Subgroup
- Results about generated normal subgroups can be found here.