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 $\text{K}$