# Definition:Generated Normal Subgroup

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## Definition

Let $G$ be a group.

Let $S \subset G$ 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 $S$:

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

### Definition 3

The **normal subgroup generated by $S$**, denoted $\gen {S^G}$, is the smallest normal subgroup of $G$ containing $S$:

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

## Also known as

The **generated normal subgroup** is also known as the **conjugate closure** or **normal closure**.