Definition:Group Presentation

Let $$G$$ be a group.

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.

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.