Presentation of a group

In mathematics, a presentation is one method of specifying a group. A presentation of a group G comprises 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

⟨

S

∣

R

⟩

.

{\displaystyle \langle S\mid R\rangle .}

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

⟨

a

∣

a

n

=

1

⟩

,

{\displaystyle \langle a\mid a^{n}=1\rangle ,}

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

⟨

a

∣

a

n

⟩

,

{\displaystyle \langle a\mid a^{n}\rangle ,}

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.

Background

A free group on a set S is a group where each element can be uniquely described as a finite length product of the form:

s

1

a

1

s

2

a

2

⋯

s

n

a

n

{\displaystyle s_{1}^{a_{1}}s_{2}^{a_{2}}\cdots s_{n}^{a_{n}}}

where the si are elements of S, adjacent si are distinct, and ai are non-zero integers (but n may be zero). In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse.