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

by means of a presentation of generators of relations among those generators. In symbols:

. But here there is a tacit assumption is the "freest" such group as clearly the relations are satisfied in any homomorphic image. One way of being able to eliminate this tacit assumption is by specifying that certain words, called the set of irrelations

Formal definition

has absolute presentation

provided that:

1. has presentation

2. Given any homomorphism is isomorphic.

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

1. is a non-trivial normal subgroup

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

namely:

1. and

. 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 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.

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:

?"

that "forces" any homomorphism preserving that specification to be an embedding. An absolute presentation does precisely this.

See also

• Presentation of a group

References