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