Absolute presentation of a group

Example

The cyclic group of order 8 has the presentation

${\displaystyle ⟨a\mid a^8=1⟩.}$

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

${\displaystyle ⟨a\mid a^4=1⟩}$

${\displaystyle ⟨a\mid a^2=1⟩}$ and

${\displaystyle ⟨a\mid a=1⟩.}$

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

${\displaystyle ⟨a\mid a^8=1,a^4\ne 1⟩.}$

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:

${\displaystyle ⟨a\mid a^8=1,a^2\ne 1⟩}$

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

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 ${\displaystyle G=⟨x_1,x_2\mid R⟩}$ can be embedded in the algebraically closed group ${\displaystyle G^{\ast }}$ then given another algebraically closed group ${\displaystyle H^{\ast }}$, we can ask "Can ${\displaystyle G}$ be embedded in ${\displaystyle H^{\ast }}$?"

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