Generating Set

Let $G$ be a group and let $S\subseteq G$. One says that $S$ generates $G$ (or that $S$ is a generating set) if the smallest subgroup of $G$ containing $S$ is all of $G$, i.e.

where $\langle S\rangle$ denotes the subgroup generated by $S$. Equivalently, every element of $G$ can be written as a finite product of elements of $S$ and their inverses.

Generating sets are the input data for presentations , and finiteness of generating sets is a basic measure of complexity.

Examples:

• $\mathbb{Z}$ is generated by $\{1\}$ under addition.
• $C_n=\langle a\rangle$ is generated by any element of order $n$.
• $S_3$ is generated by $\{(12),(123)\}$ (many other generating sets exist).