In mathematics, an absolute presentation is one method of defining a group.
by means of a presentation, one specifies a set
is the group generated by the set
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
, called the set of irrelations, such that
To define an absolute presentation of a group
of relations and irrelations among those generators.
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 (in the usual sense of the word) is sometimes referred to as a relative presentation, which is an instance of a retronym.
The cyclic group of order 8 has the presentation But, up to isomorphism there are three more groups that "satisfy" the relation
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.
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
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
can be embedded in the algebraically closed group
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.
that "forces" any homomorphism preserving that specification to be an embedding.