Absolute presentation of a group explained

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

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:

G has presentation \langle S \mid R\rangle. h:G → 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 ≠ \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} ≠ 1 . So an absolute presentation for the cyclic group of order 8 is:\langle a \mid a ^{8} = 1 , a ^{4} ≠ 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} ≠ 1 \rangle

Is not an absolute presentation for the cyclic group of order 8 because the irrelation

a ^{2} ≠ 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

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 → 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.

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.