Definition:Generator of Subgroup

Definition
Let $\struct {G, \circ}$ be a group.

Let $S \subseteq G$.

Then $S$ is a generator of $G$, denoted $G = \gen S$, $G$ is the subgroup generated by $S$.

Also denoted as
If $S$ is a singleton, that is: $S = \set x$, then we can (and usually do) write $G = \gen x$ instead of $G = \gen {\set x}$.

Also known as
This is also voiced:
 * $S$ is a generator of $\struct  {H, \circ}$
 * $S$ generates $\struct {H, \circ}$
 * $\struct {H, \circ}$ is the subgroup of $\struct {G, \circ}$ generated by $S$.

Some sources refer to such an $S$ as a set of generators of $H$, but this terminology is misleading, as it can be interpreted to mean that each of the elements of $S$ is itself a generator of $H$ independently of the other elements.

Other sources use the term generating set, which is less ambiguous.

Some sources use the notation $\operatorname{gp} \set S$ for the group generated by $S$.

Also see

 * Definition:Generated Subgroup


 * Existence of Unique Subgroup Generated by Subset
 * Set of Words Generates Group