Presentation of a group

thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that do 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.

This point of view is particularly common in the field of combinatorial group theory.

The following table lists some examples of presentations for commonly studied groups. Note that in each case there are many other presentations that are possible. The presentation listed is not necessarily the most efficient one possible.

One may take the elements of the group for generators and the Cayley table for relations.

Further, some properties of this graph (the coarse geometry) are intrinsic, meaning independent of choice of generators.