Formal Definition
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:
- has presentation
- 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. 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.
Read more about this topic: Absolute Presentation Of A Group
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