Presentation of a group

Symbol-free definition
A presentation of a group is the following data:


 * A set of elements in the group that generate the group (that is, a generating set of the group)
 * A set of words in terms of these elements, that simplify to the identity in the group (that is, a set of relations among the elements) with the property that a word in the generators simplifies to the identity if and only if it can be expressed formally as a product of conjugates of these words and their inverses

Another way of defining a presentation of a group is as follows:


 * A quotient map from a free group to the given group (the images of free generators of the generating set denote generators of the given group).
 * A set of elements in the free group whose normal closure is the kernel of the quotient map. These elements play the role of relations.

Definition with symbols
A presentation of a group is a description of the form:

$$G := \langle X \mid R \rangle$$

where $$X$$ is a set of elements (that can be thought of as generators) and $$R$$ is a set of words in those elements that evaluate to the identity in $$G$$, such that if we take the free group on the set $$X$$, then the kernel of the natural homomorphism from that to $$G$$ is the normal closure of the subgroup generated by $$R$$.

Sometimes, instead of writing the elements of $$R$$ as words, we write them as equations. Here, the corresponding word to an equation can be taken as the left hand side times the inverse of the right hand side.

Examples
In the examples below, we reserve the letter $$e$$ for the identity element, hence do not use this letter as a generator. Often, people use $$1$$ instead of $$e$$ to avoid this confusion.

(Note that chain equalities mean that each of the equalities in the chain is a relation. It suffices to take all adjacent-pair equalities).

Multiplication table presentation
In the multiplication table presentation of a group, we take the generating set as the set of all elements of the group and the set of relations as all the multiplication relations. Clearly, these relations are sufficient to determine the group.

For a finite group of order $$n$$, this has $$n$$ generators and $$n^3$$ relations, each relation being a word of length three (indicating the two elements being multiplied, multiplied by the inverse of their product).

Finite presentation
A finite presentation of a group is a presentation where both the generating set and the set of relations is finite. A group that possesses a finite presentation is termed a finitely presented group.

A related notion is that of recursive presentation and recursively presented group.

Balanced presentation
A balanced presentation is one where the number of generators equals the number of relations.

More generally, the deficiency of a presentation measures the difference between the number of generators and the number of relators.

Effect of group operations
We denote the input groups by $$G_1$$ and $$G_2$$, their number of generators by $$g_1$$ and $$g_2$$ respectively, and their number of relators by $$r_1$$ and $$r_2$$ respectively.

Manipulating presentations
There are various techniques of manipulating presentations of a group to obtain new presentations, and further, to use presentations of a group to obtain presentations of a subgroup.