Equivalence of Definitions of Generated Normal Subgroup

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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 subgroup generated by the set of conjugates of $S$:

$\gen {S^G} = \set {g^{−1}sg: g \in G, s \in S}$

Definition 3

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

$(1)$ is equivalent to $(2)$




$(1)$ is equivalent to $(3)$

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 $S$:

$H \in \mathbb S$

By Intersection is Subset:

$\ds \bigcap \mathbb S \subseteq H$


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.

$\blacksquare$