# Definition:Generated Normal Subgroup/Definition 3

## Definition

Let $G$ be a group.

Let $S \subseteq G$ be a subset.

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

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