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.

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

• The free group with $n$ generators has a presentation with $n$ generators and no relations:

$F_{n}:=\langle a_{1},a_{2},\dots ,a_{n}\mid \rangle$

In particular, the group of integers has a presentation with one generator and no relations.

• The free abelian group with $n$ generators has a presentation with $n$ generators and ${\binom {n}{2}}$ relations given by the commutation relations between all pairs of generators. For instance, the free abelian group with three generators (isomorphic to $\mathbb {Z} ^{3}$) is given as:

$\langle a,b,c\mid ab=ba,bc=cb,ac=ca\rangle$

Converting equations to words, we obtain the other way of writing this presentation:

$\langle a,b,c\mid aba^{-1}b^{-1},bcb^{-1}c^{-1},aca^{-1}c^{-1}\rangle$

• The finite cyclic group of order $n$, isomorphic to the group of integers modulo n $mathbb{Z}/n\mathbb {Z}$, has the presentation:

$\langle a\mid a^{n}=e\rangle$

In other words, $a^{n}$ is the identity element.

• The dihedral group of degree $n$ (order $2n$) has the presentation:

$\langle a,x\mid a^{n}=e,x^{2}=e,xax=a^{-1}\rangle$

Here, $e$ (often written as $1$ to not confuse with presentation letters) is the identity element).

• The symmetric group of degree four has the presentation:

$\langle s_{1},s_{2},s_{3}\mid s_{1}^{2}=s_{2}^{2}=s_{3}^{2}=e,(s_{1}s_{2})^{3}=(s_{2}s_{3})^{3}=e,s_{1}s_{3}=s_{3}s_{1}\rangle$

More generally, symmetric groups on finite sets are Coxeter groups.

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

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.

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.

Operation	Presentation of output in terms of presentations of inputs	Number of generators of output	Number of generators and relations of output
external direct product $G_{1}\times G_{2}$	We take the (disjoint) union of the generating sets and the (disjoint) union of the relation sets, and add in relations stating that every generator of $G_{1}$ commutes with every generator of $G_{2}$	$g_{1}+g_{2}$ generators, $r_{1}+r_{2}+g_{1}g_{2}$ relations
external semidirect product $G_{1}\rtimes G_{2}$ ($G_{2}$ acts on $G_{1}$)	We take the (disjoint) union of the generating sets and the (disjoint) union of the relation sets, and add in an action relation for the action of every generator of $G_{2}$ on every generator of $G_{1}$, as well as (in the infinite case) an action relation for the action of the inverse of every generator of $G_{2}$ on every element of $G_{1}$	$g_{1}+g_{2}$ generators, $r_{1}+r_{2}+2g_{1}g_{2}$ relations. In the finite case, suffices to have $r_{1}+r_{2}+g_{1}g_{2}$ relations
external wreath product $G_{1}\wr G_{2}$, with the acting group $G_{2}$ finite of order $h$	We take the (disjoint) union of the generating sets, the (disjoint) union of the relations, and, for any two (possibly equal) generators of $G_{1}$ and every element of $G_{2}$, a commutativity relation between the first generator for $G_{1}$ and the conjugate by the element of $G_{2}$ of the second generator	$g_{1}+g_{2}$ generators, $r_{1}+r_{2}+g_{1}^{2}h$ relations
external free product $G_{1}*G_{2}$	We take the (disjoint) union of the generators, and the (disjoint) union of the relations	$g_{1}+g_{2}$ generators, $r_{1}+r_{2}$ relations

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.

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F05

External links

Definition links

• Wikipedia page
• Planetmath page
• Mathworld page