# 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

In the examples below, we reserve the letter $e$ for the identity element, hence do not use this letter as a generator. Often, people use $1$ instead of $e$ to avoid this confusion.

Example group Presentation used Why the presentation works
free group with $n$ generators presentation with $n$ generators and no relations: $F_n := \langle a_1, a_2, \dots, a_n \mid \rangle$ The no relations means that the normal closure of the relation set is the trivial subgroup, so we get the quotient of the free group on $n$ generators by the trivial subgroup, thus also giving the free group on $n$ generators.
group of integers presentation with one generator and no relation: $\langle a \rangle$ special case of free group, with 1 generator.
free abelian group of rank $n$ 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:$\langle a,b,c \mid aba^{-1}b^{-1}, bcb^{-1}c^{-1}, aca^{-1}c^{-1} \rangle$
once we impose the relation that any two generators commute, this forces all elements in the group to commute with each other. Equivalently, it is the quotient of the free group on $n$ generators by its derived subgroup, i.e., the abelianization.
finite cyclic group of order $n$, isomorphic to the group of integers modulo n $\mathbb{Z}/n\mathbb{Z}$, has the presentation:$\langle a \mid a^n = e\rangle$ We start with the free group on 1 generator and quotient out by the $n^{th}$ powers. In additive notation, we are quotienting $\mathbb{Z}$ by the subgroup $n\mathbb{Z}$.
The dihedral group of degree $n$ (order $2n$) $\langle a,x \mid a^n = e, x^2 = e, xax^{-1} = a^{-1} \rangle$
Note: Since $x = x^{-1}$, the third relation can also be written as $xax = a^{-1}$
See presentation of semidirect product is disjoint union of presentations plus action by conjugation relations.

(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.

For a finite group of order $n$, this has $n$ generators and $n^3$ relations, each relation being a word of length three (indicating the two elements being multiplied, multiplied by the inverse of their product).

### 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 Proof
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, $r_1 + r_2 + g_1g_2$ relations [[presentation of direct product is disjoint union of presentations plus commutation 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, $r_1 + r_2 + 2g_1g_2$ relations. In the finite case, suffices to have $r_1 + r_2 + g_1g_2$ relations [[presentation of semidirect product is disjoint union of presentations plus action by conjugation 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, $r_1 + r_2 + g_1^2h$ relations Follows from the result for presentation of semidirect product
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 presentation of free product is disjoint union of presentations

## 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