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In mathematics, an absolute presentation is one method of defining a group.
Recall that to define a group G {\displaystyle G\ } by means of a presentation, one specifies a set S {\displaystyle S\ } of generators so that every element of the group can be written as a product of some of these generators, and a set R {\displaystyle R\ } of relations among those generators. In symbols:
Informally G {\displaystyle G\ } is the group generated by the set S {\displaystyle S\ } such that r = 1 {\displaystyle r=1\ } for all r ∈ R {\displaystyle r\in R}. But here there is a tacit assumption that G {\displaystyle G\ } is the "freest" such group as clearly the relations are satisfied in any homomorphic image of G {\displaystyle G\ }. One way of being able to eliminate this tacit assumption is by specifying that certain words in S {\displaystyle S\ } should not be equal to 1. {\displaystyle 1.} That is we specify a set I {\displaystyle I\ } , called the set of irrelations, such that i ≠ 1 {\displaystyle i\neq 1\ } for all i ∈ I {\displaystyle i\in I}.