# Equivalence of Definitions of Generated Subgroup

## Theorem

The following definitions of the concept of Generated Subgroup are equivalent:

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 $S \cup S^{-1}$.

## Proof

### $(1)$ implies $(2)$

Let $\gen S$ be a generated subgroup by definition 1.

Then by definition:

$\gen S$ is the smallest subgroup containing $S$

Thus $\gen S$ is a generated subgroup by definition 2.

$\Box$

### $(2)$ implies $(1)$

Let $\gen S$ be a generated subgroup by definition 2.

Then by definition:

$\gen S$ is the intersection of all subgroups of $G$ containing $S$.

Thus $\gen S$ is a generated subgroup by definition 1.

$\blacksquare$