# 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

### $(1)$ is equivalent to $(2)$

Let $H$ be the smallest subgroup containing $S$.

Let $\mathbb S$ be the set of subgroups containing $S$.

To show the equivalence of the two definitions, we need to show that $H = \bigcap \mathbb S$.

Since $H$ is a subgroup containing $S$:

- $H \in \mathbb S$

- $\bigcap \mathbb S \subseteq H$

On the other hand, by Intersection of Subgroups is Subgroup:

- $\bigcap \mathbb S$ is a subgroup containing $S$.

Since $H$ be the smallest subgroup containing $S$:

- $H \subseteq \bigcap \mathbb S$

By definition of set equality:

- $H = \bigcap \mathbb S$

Hence the result.

$\Box$

### $(1)$ is equivalent to $(3)$

This is shown in Set of Words Generates Group.

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $5$: Subgroups: Exercise $12$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: Exercise $\text{K}$