In mathematics, one method of defining a group is by an absolute presentation.
Contents
Recall that to define a group
Informally
Formal Definition
To define an absolute presentation of a group
provided that:
-
G ⟨ S ∣ R ⟩ . - Given any homomorphism
h : G → H such that the irrelationsI h ( G ) G h ( G )
A more algebraic, but equivalent, way of stating condition 2 is:
2a. ifRemark: 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. The term seems rather strange as one may well ask "relative to what?" and the only justification seems to be that relative is habitually used as an antonym to absolute.
Example
The cyclic group of order 8 has the presentation
But, up to isomorphism there are three more groups that "satisfy" the relation
However none of these satisfy the irrelation
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
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
Suppose we know that a 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