Presentation of a group

Common card · No. 497696

In mathematics, a presentation is one method of specifying a group. A presentation of a group G comprises a set S of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set R of relations among those generators. We then say G has presentation ⟨ S ∣ R ⟩ . {\displaystyle \langle S\mid R\rangle .} Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the presentation ⟨ a ∣ a n = 1 ⟩ , {\displaystyle \langle a\mid...

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