# Definition:Generator of Group

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## Definition

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

Let $S \subseteq G$.

Then **$S$ is a generator of $G$**, denoted $G = \gen S$, if and only if $G$ 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 $\struct {G, \circ}$ - $S$
**generates**$\struct {G, \circ}$

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

- Results about
**generators of groups**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**generator**:**2.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**generator**:**2.**