Presentation of a group

Definition

Let $S$ be a set, let $F_S$ be the free group on $S$, and let $R \subseteq F_S$. Let $N$ be the smallest normal subgroup of $F_S$ that contains $R$, the normal closure of $R$. The group with presentation $\langle S \mid R \rangle$ is the quotient group $F_S / N$. A group $G$ has the presentation $\langle S \mid R \rangle$ when $G$ is isomorphic to $F_S / N$.

Remarks

The elements of $S$ are the generators and the elements of $R$ are the relators; a relation $x = y$ stands for the relator $y^{-1} x$. The cyclic group of order $n$ has the presentation $\langle a \mid a^n \rangle$, and the dihedral group of order $16$ has the presentation $\langle r, f \mid r^8, f^2, (r f)^2 \rangle$. Each group has a presentation, and each finite group has a presentation with $S$ and $R$ finite. A group is finitely presented when it has a presentation with $S$ and $R$ finite.

Explanations

L1 human

A presentation lists generators and the rules that they obey. The group is the largest group that these generators make, subject only to these rules. Presentations are often the shortest way to describe a group.