# Definition:Group Presentation

## Informal definition

Let $G$ be a group.

A **(group) presentation for $G$** is a definition in the form:

- $G = \gen {a_1, a_2, \ldots, a_n: w_1 = e, w_2 = e, \ldots, w_k = e}$

where:

- $a_1, a_2, \ldots, a_n$ is a list of generators of $G$
- $w_1 = e, w_2 = e, \ldots, w_k = e$ is a list of equations specifying
*relations*between powers of these generators.

## Definition

Let $G$ be a group.

A **presentation for $G$** is a triple $\tuple {S, R, f}$ where:

- $S$ is a set
- $R$ is a set of relations on $S$
- $f: \gen {S \mid R} \to G$ is a group isomorphism from the group defined by $\struct {S, R}$.

## Relations

The standard form of a **relation** in a **group presentation** is:

- $w = e$

where $w$ is a word in the group.

## Comment

We defined the notation $\gen S$ to be the group generated by $S$ where $S \subseteq G$. This is the subgroup of $G$ which is generated by $S$.

However, here we have taken the concept of $\gen S$ out of the context of the group of which $S$ is a subset, and used it to define a group from first principles.

When $S \subseteq G$, where $G$ and its elements are well-defined, the relations between the elements of $G$ are all documented and understood. However, when using $\gen S$ to *define* a group, there *are* no relations between the elements *until we define them*. In fact, in a truly abstract sense, neither are the elements of $S$ defined in the context of $G$ except from their membership of $S$ and their relationships between each other.

## Also see

- Results about
**group presentations**can be found here.

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $4$: Subgroups: Example $4.10$ - 2000: Pierre A. Grillet:
*Abstract Algebra*: $\S \text{I}.7$: Presentations