Equivalence of Definitions of Generated Normal Subgroup
Theorem
The following definitions of the concept of Generated Normal Subgroup are equivalent:
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 $S$:
- $\gen {S^G} = \gen {x S x^{-1}: x \in G}$
Proof
Let $H$ be the smallest normal subgroup containing $S$.
Let $\mathbb S$ be the set of normal subgroups containing $S$.
To show the equivalence of the two definitions, we need to show that $\ds H = \bigcap \mathbb S$.
Since $H$ is a normal subgroup containing $S$:
- $H \in \mathbb S$
- $\ds \bigcap \mathbb S \subseteq H$
On the other hand, by Intersection of Normal Subgroups is Normal:
- $\ds \bigcap \mathbb S$ is a normal subgroup containing $S$.
Since $H$ be the smallest normal subgroup containing $S$:
- $\ds H \subseteq \bigcap \mathbb S$
By definition of set equality:
- $\ds H = \bigcap \mathbb S$
Hence the result.
$\blacksquare$