Definition:Generator of Group

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 $G = \gen x$ for the group generated by $\set x$ rather than $G = \gen {\set x}$.

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 $\struct {G, \circ}$

$S$ 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.

Also see

• Definition:Generated Subgroup

• Definition:Generator of Cyclic Group
• Definition:Generator of Subgroup

• Set of Words Generates Group

• 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.