linguistic:translation, information:extraction
Presentation

A presentation of a group is a set of elements that generate the group (meaning that all group elements can be expressed as products of finitely many generators), together with "relators", products of generators that give the identity element. The elements of a group defined in this way are equivalence classes of sequences of generators and their inverses, under an equivalence relation that allows inserting or removing any relator or generator-inverse pair as a contiguous subsequence. The free abelian group with basis B {\displaystyle B} has a presentation in which the generators are the elements of B {\displaystyle B}, and the relators are the commutators of pairs of elements of B {\displaystyle B}. Here, the commutator of two elements x {\displaystyle x} and y {\displaystyle y} is the product x − 1 y − 1 x y {\displaystyle x^{-1}y^{-1}xy}; setting this product to the identity causes x y {\displaystyle xy} to equal y x {\displaystyle yx}, so that x {\displaystyle x} and y {\displaystyle y} commute. More generally, if all pairs of generators commute, then all pairs of products of generators also commute. Therefore, the group generated by this presentation is abelian, and the relators of the presentation form a minimal set of relators needed to ensure that it is abelian.