Subgroup Generated by Subgroup

Theorem

Let $G$ be a group.

Let $H \le G$ be a subgroup of $G$.

Then:

$H = \gen H$

where $\gen H$ denotes the subgroup generated by $H$.

Proof

By definition of generated subgroup, $\gen H$ is the smallest subgroup of $H$ containing $H$.

Hence the result.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.3$. Subgroup generated by a subset: Example $97$