Definition:Group Defined by Presentation

Definition
Let $S$ be a set.

Let $R$ be a set of group words on $S$.

Let $F_S$ be the group of reduced formal words on $S$.

Let $\operatorname{red}(R)$ be the set of reduced forms of elements of $R$.

Let $N$ be the normal subgroup generated by $\operatorname{red}(R)$ in $F$.

The group defined by the presentation $\langle S \mid R \rangle$ is the quotient group $F_S/N$.

Also see

 * Definition:Presentation of Group
 * Definition:Free Group