Generators of a group (reviewed)

If $G$ is a group and $S$ is a subset of $G$, then $S$ is a set of generators if the smallest subgroup of $G$ containing $S$ equals $G$.

Equivalently, $S$ generates $G$ if \[ G=\bigcap_{S\subseteq H\leq G} H \,.\]

The automorphism group of $G$ acts on such $S$, and we say $S$ and $S'$ are equivalent if they are related by this action.