Absolute presentation of a group

In mathematics, an absolute presentation is one method of defining a group. ^[1]

Recall that to define a group ${\displaystyle G\ }$ by means of a presentation, one specifies a set ${\displaystyle S\ }$ of generators so that every element of the group can be written as a product of some of these generators, and a set ${\displaystyle R\ }$ of relations among those generators. In symbols:

${\displaystyle G\simeq \langle S\mid R\rangle .}$

Informally ${\displaystyle G\ }$ is the group generated by the set ${\displaystyle S\ }$ such that ${\displaystyle r=1\ }$ for all ${\displaystyle r\in R}$. But here there is a tacit assumption that ${\displaystyle G\ }$ is the "freest" such group as clearly the relations are satisfied in any homomorphic image of ${\displaystyle G\ }$. One way of being able to eliminate this tacit assumption is by specifying that certain words in ${\displaystyle S\ }$ should not be equal to ${\displaystyle 1.}$ That is we specify a set ${\displaystyle I\ }$, called the set of irrelations, such that ${\displaystyle i\neq 1\ }$ for all ${\displaystyle i\in I}$.

Formal definition

To define an absolute presentation of a group ${\displaystyle G}$ one specifies a set ${\displaystyle S}$ of generators, a set ${\displaystyle R}$ of relations among those generators and a set ${\displaystyle I}$ of irrelations among those generators. We then say ${\displaystyle G}$ has absolute presentation

${\displaystyle \langle S\mid R,I\rangle .}$

provided that:

1. ${\displaystyle G}$ has presentation ${\displaystyle \langle S\mid R\rangle .}$
2. Given any homomorphism ${\displaystyle h:G\rightarrow H}$ such that the irrelations ${\displaystyle I\ }$ are satisfied in ${\displaystyle h(G)}$, ${\displaystyle G}$ is isomorphic to ${\displaystyle h(G)}$.

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

2a. If ${\displaystyle N\triangleleft G}$ is a non-trivial normal subgroup of ${\displaystyle G}$ then ${\displaystyle I\cap N\neq \left\{1\right\}.}$

Remark: The concept of an absolute presentation has been fruitful in fields such as algebraically closed groups 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

${\displaystyle \langle a\mid a^{8}=1\rangle .}$

But, up to isomorphism there are three more groups that "satisfy" the relation ${\displaystyle a^{8}=1\,}$ namely:

${\displaystyle \langle a\mid a^{4}=1\rangle }$

${\displaystyle \langle a\mid a^{2}=1\rangle }$ and

${\displaystyle \langle a\mid a=1\rangle .}$

However none of these satisfy the irrelation ${\displaystyle a^{4}\neq 1}$. So an absolute presentation for the cyclic group of order 8 is:

${\displaystyle \langle a\mid a^{8}=1,a^{4}\neq 1\rangle .}$

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:

${\displaystyle \langle a\mid a^{8}=1,a^{2}\neq 1\rangle }$

Is not an absolute presentation for the cyclic group of order 8 because the irrelation ${\displaystyle a^{2}\neq 1}$ is satisfied in the cyclic group of order 4.

Background

The notion of an absolute presentation arises from Bernhard Neumann's study of the isomorphism problem for algebraically closed groups. ^[1]

A common strategy for considering whether two groups ${\displaystyle G\,}$ and ${\displaystyle H\,}$ 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 recursively presented and so it is impossible to compare their presentations. Neumann considered the following alternative strategy:

Suppose we know that a group ${\displaystyle G\,}$ with finite presentation ${\displaystyle G=\langle x_{1},x_{2}\mid R\rangle }$ can be embedded in the algebraically closed group ${\displaystyle G^{*}\,}$ then given another algebraically closed group ${\displaystyle H^{*}\,}$, we can ask "Can ${\displaystyle G\,}$ be embedded in ${\displaystyle H^{*}\,}$?"

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 ${\displaystyle h:G\rightarrow H^{*}}$, this homomorphism need not be an embedding. What is needed is a specification for ${\displaystyle G^{*}\,}$ that "forces" any homomorphism preserving that specification to be an embedding. An absolute presentation does precisely this.

References

1. 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.