Equivalence of Definitions of Generated Subgroup

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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.

$\blacksquare$


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