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 $S$:

$H \in \mathbb S$

By Intersection is Subset:

$\bigcap \mathbb S \subseteq H$

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 $\text{K}$