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 $S$.
Where $\map P x$ is a propositional function, the notation:
- $\gen {x \in S: \map P x}$
can be seen for:
- $\gen {\set {x \in S: \map P x} }$
which is no more than notation of convenience.
Also known as
The expression $\struct {G, \circ} = \gen S$ can be voiced as:
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
- 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.