Definition:Group Presentation

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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