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 $G = \gen x$ 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:

$S$ 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.

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.

Also see

• Definition:Generated Subgroup

• Existence of Unique Subgroup Generated by Subset
• Set of Words Generates Group

• Results about generators of groups can be found here.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.3$. Subgroup generated by a subset
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$
• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35 \epsilon$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms

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• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$