# Equivalence of Definitions of Generated Subgroup

## Contents

## 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:

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.

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $5$: Subgroups: Exercise $12$