Definition:Generator of a Group

Definition

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

Let $S \subseteq G$.

The subgroup of $\left({G, \circ}\right)$ generated by $S$ is the smallest subgroup $H$ of $G$ containing $S$.

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

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 $H = \left\langle {\left\{{x}\right\}}\right\rangle$.

This subgroup is proven to exist by Generator of a Group.

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 5.3$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.9$