Definition:Group Presentation

Informal definition
Let $G$ be a group.

Informally, a presentation for $G$ is a definition in the form:


 * $G = \left \langle {a_1, a_2, \ldots, a_n: w_1 = e, w_2 = e, \ldots, w_k = e} \right \rangle$

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 $(S, R, f)$ where:
 * $S$ is a set
 * $R$ is a set of relations on $S$
 * $f : \langle S \mid R \rangle \to G$ is a group isomorphism from the group defined by $(S, R)$.

Relations
We need to define, in this context, what is meant by the term relation in the above.

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 $\left \langle {S} \right \rangle$ 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 $\left \langle {S} \right \rangle$ 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 $\left \langle {S} \right \rangle$ 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.