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 $\struct {S, R}$.

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.