\begin{definition}[Definition:Group Defined by Presentation]
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 $F_S / N$.
\end{definition}