Presentation of a group

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a presentation is one method of specifying a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>. A presentation of a group G comprises a <a href="/facts/Generating_set_of_a_group/B9q5lnFY">set S of generators</a>—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 <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> to the <a href="/facts/Quotient_group/HlwpHtjb">quotient</a> of a <a href="/facts/Free_group/zPo2NIGb">free group</a> on S by the <a href="/facts/Normal_closure_(group_theory)/k2lJahCW">normal subgroup generated by</a> the relations R.
As a simple example, the <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> of order n has the presentation

⟨
        a
        ∣
        
          a
          
            n
          
        
        =
        1
        ⟩
        ,
      
    
    {\displaystyle \langle a\mid a^{n}=1\rangle ,}

where 1 is the group identity. This may be written equivalently as

⟨
        a
        ∣
        
          a
          
            n
          
        
        ⟩
        ,
      
    
    {\displaystyle \langle a\mid a^{n}\rangle ,}

thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that do include an equals sign.
Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group.
A closely related but different concept is that of an <a href="/facts/Absolute_presentation_of_a_group/6bSUq62D">absolute presentation of a group</a>.

Presentation of a group open-in-new

Presentation of a group