Definition:Generator of Subgroup

Definition
Let $\left({G, \circ}\right)$ be a group.

Let $S \subseteq G$.

Let $H$ be the smallest subgroup of $G$ such that $S \subseteq H$.

Then:
 * $S$ is a generator of $\left({H, \circ}\right)$
 * $S$ generates $\left({H, \circ}\right)$
 * $\left({H, \circ}\right)$ is the subgroup of $\left({G, \circ}\right)$ generated by $S$.

This is written $H = \left\langle {S} \right\rangle$.

If $S$ is a singleton, i.e. $S = \left\{{x}\right\}$, then we can (and usually do) write $H = \left\langle {x}\right\rangle$ for $H = \left\langle {\left\{{x}\right\}}\right\rangle$.

Also known as
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 $\operatorname{gp} \left\{{S}\right\}$ for the group generated by $S$.

Also see

 * Definition:Generator


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