In mathematics, an absolutepresentation is one method of defining a group. 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 homomorphicimage 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.
To define an absolute presentation of a group one specifies a set of generators, a set of relations among those generators and a set of irrelations among those generators. We then say has absolute presentation provided that:
A more algebraic, but equivalent, way of stating condition 2 is: 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 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 But, up to isomorphism there are three more groups that "satisfy" the relation namely: 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
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 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 enoughinformation 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.
