Absolute presentation of a group

In mathematics, an absolute presentation is one method of defining a group.^[1]

Recall that to define a group ${\displaystyle G}$ by means of a presentation, one specifies a set ${\displaystyle S}$ of generators so that every element of the group can be written as a product of some of these generators, and a set ${\displaystyle R}$ of relations among those generators. In symbols:

${\displaystyle G\simeq \langle S\mid R\rangle .}$

Informally ${\displaystyle G}$ is the group generated by the set ${\displaystyle S}$ such that ${\displaystyle r=1}$ for all ${\displaystyle r\in R}$. But here there is a tacit assumption that ${\displaystyle G}$ is the "freest" such group as clearly the relations are satisfied in any homomorphic image of ${\displaystyle G}$. One way of being able to eliminate this tacit assumption is by specifying that certain words in ${\displaystyle S}$ should not be equal to ${\displaystyle 1.}$ That is we specify a set ${\displaystyle I}$, called the set of irrelations, such that ${\displaystyle i\neq 1}$ for all ${\displaystyle i\in I.}$