Absolute presentation of a group

In mathematics, one method of defining a group is by an absolute presentation.

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:

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 ${\displaystyle i\in I}$.

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