View source for Absolute presentation of a group

In [[mathematics]], one method of defining a [[group (mathematics)|group]] is by an '''absolute presentation'''.<ref name="neumann">B. Neumann, ''The isomorphism problem for algebraically closed groups,'' in: Word Problems, Decision Problems, and the Burnside Problem in Group Theory, Amsterdam-London (1973), pp. 553–562.</ref>

Recall that to define a group <math>G\ </math> by means of a [[presentation of a group|presentation]], one specifies a set <math>S\ </math> of '''[[generating set of a group|generators]]''' so that every element of the group can be written as a product of some of these generators, and a set <math>R\ </math> of '''relations''' among those generators. In symbols:

:<math>G \simeq \langle S \mid R \rangle.</math>

Informally  <math>G\ </math> is the group generated by the set <math>S\ </math> such that <math>r = 1\ </math> for all <math>r \in R</math>. But here there is a [[tacit assumption]] that <math>G\ </math> is the "freest" such group as clearly the relations are satisfied in any [[homomorphism|homomorphic]] image of <math>G\ </math>. One way of being able to eliminate this tacit assumption is by specifying that certain words in <math>S\ </math> should not be equal to <math>1.</math> That is we specify a set <math>I\ </math>, called the set of '''irrelations''', such that <math>i\ne 1\ </math> for all <math>i \in I</math>.

==Formal definition==

To define an absolute presentation of a group <math>G\ </math> one specifies a set <math>S\ </math> of generators, a set <math>R\ </math> of relations among those generators and a set <math>I\ </math> of irrelations among those 
generators. We then say <math>G\ </math>  has absolute presentation

:<math>\langle S \mid R, I\rangle.</math>

provided that:
# <math>G\ </math> has [[presentation of a group|presentation]] <math>\langle S \mid R\rangle.</math>
# Given any [[homomorphism]] <math>h:G\rightarrow H</math> such that the irrelations <math>I\ </math> are satisfied in <math>h(G)\ </math>, <math>G\ </math>  is [[isomorphic]] to <math>h(G)\ </math>.

A more algebraic, but equivalent, way of stating condition 2 is:

:2a. if <math>N\triangleleft G\ </math> is a non-trivial [[normal subgroup]] of <math>G </math> then <math>I\cap N\neq \left\{ 1\right\} .</math>

'''Remark:''' The concept of an absolute presentation has been fruitful in fields such as [[algebraically closed group]]s and the [[Grigorchuk topology]].
In the literature, in a context where absolute presentations are being discussed, a presentation (in the usual sense of the word) is sometimes referred to as a '''relative presentation''', which is an instance of a [[retronym]].

==Example==

The [[cyclic group]] of order ''8'' has the presentation
:<math>\langle a \mid a^8 = 1\rangle.</math>

But, up to isomorphism there are three more groups that "satisfy" the relation <math>a^8 = 1\,</math> namely:
:<math>\langle a \mid a^4 = 1\rangle</math>
:<math>\langle a \mid a^2 = 1\rangle</math> and
:<math>\langle a \mid a = 1\rangle.</math>

However none of these satisfy the irrelation <math>a^4 \neq 1</math>. So an absolute presentation for the cyclic group of order 8 is:
:<math>\langle a \mid a^8 = 1, a^4 \neq 1\rangle.</math>

It is part of the definition of an absolute presentation that the irrelations are not satisfied in any proper homomorphic image of the group. Therefore:
:<math>\langle a \mid a^8 = 1, a^2 \neq 1\rangle</math>

Is ''not'' an absolute presentation for the cyclic group of order 8 because the irrelation <math>a^2 \neq 1</math> is satisfied in the cyclic group of order 4.

== Background ==

The notion of an absolute presentation arises from [[Bernhard Neumann]]'s study of the [[Group isomorphism problem|isomorphism problem]] for [[algebraically closed group]]s.<ref name="neumann"/>

A common strategy for considering whether two groups <math>G\,</math> and <math>H\,</math> are [[isomorphic]] is to  consider whether a presentation for one might be transformed into a presentation for the other. However algebraically closed groups are neither finitely generated nor [[presentation of a group|recursively presented]] and so it is impossible to compare their presentations. Neumann considered the following alternative strategy: 

Suppose we know that a group <math>G\,</math> with finite presentation <math>G=\langle x_1,x_2 \mid R \rangle</math> can be embedded in the algebraically closed group <math>G^{*}\,</math> then given another algebraically closed group <math>H^{*}\,</math>, we can ask "Can <math>G\,</math> be embedded in <math>H^{*}\,</math>?"

It soon becomes apparent that a presentation for a group does not contain enough information to make this decision for while there may be a homomorphism <math>h:G\rightarrow H^{*}</math>, this homomorphism need not be an embedding. What is needed is a specification for <math>G^{*}\,</math> that "forces" any homomorphism preserving that specification to be an embedding. An absolute presentation does precisely this.

==References==

<references/>

[[Category:Combinatorial group theory]]