# 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 $S$**.

## Also known as

This is also voiced:

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

- 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): $\S 1.2$