202403132303

Tags : Group Theory

Presentation of a Group

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A presentation is a way to represent a graph.

It comprises of a set of generators $S$ and a set of relations $R$ among the generators and we write it as follows

$$G=⟨S|R⟩$$

This means that $G$ is in some sense the “free-est” group that follows the set of relations $R$.

Formal formal

Formally there are two ways to interpret the above statement.

• The easy way : It is a quotient on the free group generated by $S$ quotiented by the free group generated by $R$.

Example

• The group $⟨\{a,b\}|⟩$ is the free group of rank 2
• The group $⟨\{a,b|ab=ba\}⟩$. This is the free abelian group of rank 2.

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References