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Absolute presentation of a group

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

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

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

Formal definition

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To define an absolute presentation of a group one specifies a set of generators and sets and of relations and irrelations among those generators. We then say has absolute presentation

provided that:

  1. has presentation
  2. Given any homomorphism such that the irrelations are satisfied in , is isomorphic to .

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

2a. If is a non-trivial normal subgroup of then

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

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The cyclic group of order 8 has the presentation

But, up to isomorphism there are three more groups that "satisfy" the relation namely:

and

However, none of these satisfy the irrelation . So an absolute presentation for the cyclic group of order 8 is:

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:

Is not an absolute presentation for the cyclic group of order 8 because the irrelation is satisfied in the cyclic group of order 4.

Background

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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 and 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 with finite presentation can be embedded in the algebraically closed group then given another algebraically closed group , we can ask "Can be embedded in ?"

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 , this homomorphism need not be an embedding. What is needed is a specification for that "forces" any homomorphism preserving that specification to be an embedding. An absolute presentation does precisely this.

References

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  1. ^ a b 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.