Presentation of a group

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Template:Group description rule

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

External links

Definition links