Generating Set - Knowlpedia

Let $G$ be a group and let $S\subseteq G$ . The set $S$ generates $G$ , or is a generating set for $G$ , if the smallest subgroup containing $S$ is $G$ :

\langle S\rangle = G,

where $\langle S\rangle$ denotes the subgroup generated by $S$ .

Equivalent characterizations

Equivalently, every element of $G$ can be written as a finite product of elements of $S$ and their inverses.

Remarks

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).