Editing Presentation of a group (section)

== Definition ==
{{redirect|Relator}}
Let ''S'' be a set and let ''F<sub>S</sub>'' be the [[free group]] on ''S''.  Let ''R'' be a set of [[Word (group theory)|words]] on ''S'', so ''R'' naturally gives a subset of <math>F_S</math>. To form a group with presentation <math>\langle S \mid R\rangle</math>, take the quotient of <math>F_S</math> by the smallest normal subgroup that contains each element of ''R''.  (This subgroup is called the [[Normal closure (group theory)|normal closure]] ''N'' of ''R'' in <math>F_S</math>.)  The group <math>\langle S \mid R\rangle</math> is then defined as the [[quotient group]]
:<math>\langle S \mid R \rangle = F_S / N.</math>
The elements of ''S'' are called the '''generators''' of <math>\langle S \mid R\rangle</math> and the elements of ''R'' are called the '''relators'''. A group ''G'' is said to have the presentation <math>\langle S \mid R\rangle</math> if ''G'' is isomorphic to <math>\langle S \mid R\rangle</math>.<ref name = Peifer>{{cite journal|first = David|last = Peifer|title = An Introduction to Combinatorial Group Theory and the Word Problem|journal = Mathematics Magazine | volume = 70 | issue = 1 | pages = 3–10 | doi = 10.1080/0025570X.1997.11996491 | year = 1997}}</ref>

It is a common practice to write relators in the form <math>x = y</math> where ''x'' and ''y'' are words on ''S''. What this means is that <math>y^{-1}x\in R</math>. This has the intuitive meaning that the images of ''x'' and ''y'' are supposed to be equal in the quotient group. Thus, for example, ''r<sup>n</sup>'' in the list of relators is equivalent with <math>r^n=1</math>.<ref name = Peifer />

For a finite group ''G'', it is possible to build a presentation of ''G'' from the [[Cayley table|group multiplication table]], as follows. Take ''S'' to be the set elements <math>g_i</math> of ''G'' and ''R'' to be all words of the form <math>g_ig_jg_k^{-1}</math>, where <math>g_ig_j=g_k</math> is an entry in the multiplication table.

=== Alternate definition ===
The definition of group presentation may alternatively be recast in terms of [[equivalence class]]es of words on the alphabet <math>S \cup S^{-1}</math>.  In this perspective, we declare two words to be equivalent if it is possible to get from one to the other by a sequence of moves, where each move consists of adding or removing a consecutive pair <math>x x^{-1}</math> or <math>x^{-1} x</math> for some {{mvar|x}} in {{mvar|S}}, or by adding or removing a consecutive copy of a relator.  The group elements are the equivalence classes, and the group operation is concatenation.<ref name = Peifer />

This point of view is particularly common in the field of [[combinatorial group theory]].