Presentation of a group

Definition

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.

Particular cases

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.

Finite presentation

Further information: 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

Further information: 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.

Related group properties

Refer Category: Properties of group presentations and Category: Presentation-based group properties

Relation with other group description rules

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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.