# Definition:Generated Subgroup

## Definition

Let $G$ be a group.

Let $S \subset G$ be a subset.

### Definition 1

The subgroup generated by $S$ is the smallest subgroup containing $S$.

### Definition 2

The subgroup generated by $S$ is the intersection of all subgroups of $G$ containing $S$.

### Definition 3

Let $S^{-1}$ be the set of inverses of $S$.

The subgroup generated by $S$ is the set of words on the union $S \cup S^{-1}$.