Presentation of A Group

Presentation Of A Group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators. We then say G has presentation

Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

As a simple example, the cyclic group of order n has the presentation

where 1 is the group identity. This may be written equivalently as

since terms that don't include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that include an equals sign.

Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group.

A closely related but different concept is that of an absolute presentation of a group.