# 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 group words on $S$.

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

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

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