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 $\map {\mathrm {red} } R$ be the set of reduced forms of elements of $R$.

Let $N$ be the normal subgroup generated by $\map {\mathrm {red} } R$ in $F$.

The group defined by the presentation $\gen {S \mid R}$ is the quotient group $F_S / N$.

Also see

• Universal Property of Group defined by Presentation
• Definition:Group Presentation
• Definition:Free Group

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