Absolute presentation of a group

In mathematics, an absolute presentation is one method of defining a group.

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


 * $$G \simeq \langle S \mid R \rangle.$$

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

Formal definition
To define an absolute presentation of a group $$G$$ one specifies a set $$S$$ of generators and sets $$R$$ and $$I$$ of relations and irrelations among those generators. We then say $$G$$ has absolute presentation


 * $$\langle S \mid R, I\rangle.$$

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

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


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

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
 * $$\langle a \mid a^8 = 1\rangle.$$

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

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

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:
 * $$\langle a \mid a^8 = 1, a^2 \neq 1\rangle$$

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

A common strategy for considering whether two groups $$G$$ and $$H$$ 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 $$G$$ with finite presentation $$G=\langle x_1,x_2 \mid R \rangle$$ can be embedded in the algebraically closed group $$G^{*}$$ then given another algebraically closed group $$H^{*}$$, we can ask "Can $$G$$ be embedded in $$H^{*}$$?"

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