Group is Generated by Itself

Theorem

Let $G$ be a group.

Then:

$G = \gen G$

where $\gen G$ denotes the group generated by $G$.

Proof

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

Hence the result by Group is Subgroup of Itself.

$\blacksquare$

Sources

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