# Definition:Generator of Subgroup

## Definition

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

Let $S \subseteq G$.

Let $H$ be the subgroup generated by $S$.

Then $S$ is a generator of $H$, denoted $H = \gen S$, if and only if $H$ is the subgroup generated by $S$.

### Definition by Predicate

A generator of a subgroup can be defined by a predicate.

For example:

$\gen {x \in G: x^2 = e}$

defines the subgroup of $G$ generated by the elements of $G$ of order $2$.

## Also denoted as

If $S$ is a singleton, that is: $S = \set x$, then we can (and usually do) write $H = \gen x$ instead of $H = \gen {\set x}$.

Some sources use the notation $\operatorname {gp} \set S$ for the subgroup generated by $S$.

## Also known as

This is also voiced:

$S$ is a generator of $\struct {H, \circ}$
$S$ generates $\struct {H, \circ}$
$\struct {H, \circ}$ is the subgroup of $\struct {G, \circ}$ generated by $S$.

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.

## Examples

### Positive Odd Numbers

Let $A$ be the set of positive odd integers.

Let $\struct {\Z, +}$ be the additive group of integers.

The subgroup of $\struct {\Z, +}$ generated by $A$ is $\struct {\Z, +}$ itself.