Definition:Generated Normal Subgroup

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 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 $S$:

$\gen {S^G} = \gen {x S x^{-1}: x \in G}$

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